P. 97 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9601-9700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	elom3 9601*	A simplification of elom 7845 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
⊢ (𝐴 ∈ ω ↔ ∀𝑥(Lim 𝑥 → 𝐴 ∈ 𝑥))
Theorem	dfom4 9602*	A simplification of df-om 7843 assuming the Axiom of Infinity. (Contributed by NM, 30-May-2003.)
⊢ ω = {𝑥 ∣ ∀𝑦(Lim 𝑦 → 𝑥 ∈ 𝑦)}
Theorem	dfom5 9603	ω is the smallest limit ordinal and can be defined as such (although the Axiom of Infinity is needed to ensure that at least one limit ordinal exists). Theorem 1.23 of [Schloeder] p. 4. (Contributed by FL, 22-Feb-2011.) (Revised by Mario Carneiro, 2-Feb-2013.)
⊢ ω = ∩ {𝑥 ∣ Lim 𝑥}
Theorem	oancom 9604	Ordinal addition is not commutative. This theorem shows a counterexample. Remark in [TakeutiZaring] p. 60. (Contributed by NM, 10-Dec-2004.)
⊢ (1o +o ω) ≠ (ω +o 1o)
Theorem	isfinite 9605	A set is finite iff it is strictly dominated by the class of natural number. Theorem 42 of [Suppes] p. 151. The Axiom of Infinity is used for the forward implication. (Contributed by FL, 16-Apr-2011.)
⊢ (𝐴 ∈ Fin ↔ 𝐴 ≺ ω)
Theorem	fict 9606	A finite set is countable (weaker version of isfinite 9605). (Contributed by Thierry Arnoux, 27-Mar-2018.)
⊢ (𝐴 ∈ Fin → 𝐴 ≼ ω)
Theorem	nnsdom 9607	A natural number is strictly dominated by the set of natural numbers. Example 3 of [Enderton] p. 146. (Contributed by NM, 28-Oct-2003.)
⊢ (𝐴 ∈ ω → 𝐴 ≺ ω)
Theorem	omenps 9608	Omega is equinumerous to a proper subset of itself. Example 13.2(4) of [Eisenberg] p. 216. (Contributed by NM, 30-Jul-2003.)
⊢ ω ≈ (ω ∖ {∅})
Theorem	omensuc 9609	The set of natural numbers is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
⊢ ω ≈ suc ω
Theorem	infdifsn 9610	Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)
Theorem	infdiffi 9611	Removing a finite set from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.)
⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ Fin) → (𝐴 ∖ 𝐵) ≈ 𝐴)
Theorem	unbnn3 9612*	Any unbounded subset of natural numbers is equinumerous to the set of all natural numbers. This version of unbnn 9243 eliminates its hypothesis by assuming the Axiom of Infinity. (Contributed by NM, 4-May-2005.)
⊢ ((𝐴 ⊆ ω ∧ ∀𝑥 ∈ ω ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦) → 𝐴 ≈ ω)
Theorem	noinfep 9613*	Using the Axiom of Regularity in the form zfregfr 9558, show that there are no infinite descending ∈-chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)
⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)
2.6.3  Cantor normal form
Syntax	ccnf 9614	Extend class notation with the Cantor normal form function.
class CNF
Definition	df-cnf 9615*	Define the Cantor normal form function, which takes as input a finitely supported function from 𝑦 to 𝑥 and outputs the corresponding member of the ordinal exponential 𝑥 ↑o 𝑦. The content of the original Cantor Normal Form theorem is that for 𝑥 = ω this function is a bijection onto ω ↑o 𝑦 for any ordinal 𝑦 (or, since the function restricts naturally to different ordinals, the statement that the composite function is a bijection to On). More can be said about the function, however, and in particular it is an order isomorphism for a certain easily defined well-ordering of the finitely supported functions, which gives an alternate definition cantnffval2 9648 of this function in terms of df-oi 9463. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ CNF = (𝑥 ∈ On, 𝑦 ∈ On ↦ (𝑓 ∈ {𝑔 ∈ (𝑥 ↑m 𝑦) ∣ 𝑔 finSupp ∅} ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝑥 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
Theorem	cantnffval 9616*	The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) = (𝑓 ∈ 𝑆 ↦ ⦋OrdIso( E , (𝑓 supp ∅)) / ℎ⦌(seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (ℎ‘𝑘)) ·o (𝑓‘(ℎ‘𝑘))) +o 𝑧)), ∅)‘dom ℎ)))
Theorem	cantnfdm 9617*	The domain of the Cantor normal form function (in later lemmas we will use dom (𝐴 CNF 𝐵) to abbreviate "the set of finitely supported functions from 𝐵 to 𝐴"). (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = 𝑆)
Theorem	cantnfvalf 9618*	Lemma for cantnf 9646. The function appearing in cantnfval 9621 is unconditionally a function. (Contributed by Mario Carneiro, 20-May-2015.)
⊢ 𝐹 = seqω((𝑘 ∈ 𝐴, 𝑧 ∈ 𝐵 ↦ (𝐶 +o 𝐷)), ∅)    ⇒   ⊢ 𝐹:ω⟶On
Theorem	cantnfs 9619	Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)))
Theorem	cantnfcl 9620	Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹 ∈ 𝑆 is a finite subset of 𝐴, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω))
Theorem	cantnfval 9621*	The value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (𝐻‘dom 𝐺))
Theorem	cantnfval2 9622*	Alternate expression for the value of the Cantor normal form function. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seqω((𝑘 ∈ dom 𝐺, 𝑧 ∈ On ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)‘dom 𝐺))
Theorem	cantnfsuc 9623*	The value of the recursive function 𝐻 at a successor. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ ((𝜑 ∧ 𝐾 ∈ ω) → (𝐻‘suc 𝐾) = (((𝐴 ↑o (𝐺‘𝐾)) ·o (𝐹‘(𝐺‘𝐾))) +o (𝐻‘𝐾)))
Theorem	cantnfle 9624*	A lower bound on the CNF function. Since ((𝐴 CNF 𝐵)‘𝐹) is defined as the sum of (𝐴 ↑o 𝑥) ·o (𝐹‘𝑥) over all 𝑥 in the support of 𝐹, it is larger than any of these terms (and all other terms are zero, so we can extend the statement to all 𝐶 ∈ 𝐵 instead of just those 𝐶 in the support). (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 28-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐴 ↑o 𝐶) ·o (𝐹‘𝐶)) ⊆ ((𝐴 CNF 𝐵)‘𝐹))
Theorem	cantnflt 9625*	An upper bound on the partial sums of the CNF function. Since each term dominates all previous terms, by induction we can bound the whole sum with any exponent 𝐴 ↑o 𝐶 where 𝐶 is larger than any exponent (𝐺‘𝑥), 𝑥 ∈ 𝐾 which has been summed so far. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘))) +o 𝑧)), ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐾 ∈ suc dom 𝐺)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → (𝐺 “ 𝐾) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → (𝐻‘𝐾) ∈ (𝐴 ↑o 𝐶))
Theorem	cantnflt2 9626	An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶))
Theorem	cantnff 9627	The CNF function is a function from finitely supported functions from 𝐵 to 𝐴, to the ordinal exponential 𝐴 ↑o 𝐵. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆⟶(𝐴 ↑o 𝐵))
Theorem	cantnf0 9628	The value of the zero function. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝐵 × {∅})) = ∅)
Theorem	cantnfrescl 9629*	A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ 𝑇 = dom (𝐴 CNF 𝐷)    ⇒   ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇))
Theorem	cantnfres 9630*	The CNF function respects extensions of the domain to a larger ordinal. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝐷)    &   ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅)    &   ⊢ (𝜑 → ∅ ∈ 𝐴)    &   ⊢ 𝑇 = dom (𝐴 CNF 𝐷)    &   ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑛 ∈ 𝐵 ↦ 𝑋)) = ((𝐴 CNF 𝐷)‘(𝑛 ∈ 𝐷 ↦ 𝑋)))
Theorem	cantnfp1lem1 9631*	Lemma for cantnfp1 9634. (Contributed by Mario Carneiro, 20-Jun-2015.) (Revised by AV, 30-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ 𝑆)
Theorem	cantnfp1lem2 9632*	Lemma for cantnfp1 9634. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 30-Jun-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    &   ⊢ (𝜑 → ∅ ∈ 𝑌)    &   ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))    ⇒   ⊢ (𝜑 → dom 𝑂 = suc ∪ dom 𝑂)
Theorem	cantnfp1lem3 9633*	Lemma for cantnfp1 9634. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    &   ⊢ (𝜑 → ∅ ∈ 𝑌)    &   ⊢ 𝑂 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐹‘(𝑂‘𝑘))) +o 𝑧)), ∅)    &   ⊢ 𝐾 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝑀 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝐾‘𝑘)) ·o (𝐺‘(𝐾‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺)))
Theorem	cantnfp1 9634*	If 𝐹 is created by adding a single term (𝐹‘𝑋) = 𝑌 to 𝐺, where 𝑋 is larger than any element of the support of 𝐺, then 𝐹 is also a finitely supported function and it is assigned the value ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑧 where 𝑧 is the value of 𝐺. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 1-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐴)    &   ⊢ (𝜑 → (𝐺 supp ∅) ⊆ 𝑋)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝑆 ∧ ((𝐴 CNF 𝐵)‘𝐹) = (((𝐴 ↑o 𝑋) ·o 𝑌) +o ((𝐴 CNF 𝐵)‘𝐺))))
Theorem	oemapso 9635*	The relation 𝑇 is a strict order on 𝑆 (a corollary of wemapso2 9506). (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → 𝑇 Or 𝑆)
Theorem	oemapval 9636*	Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))))
Theorem	oemapvali 9637*	If 𝐹 < 𝐺, then there is some 𝑧 witnessing this, but we can say more and in fact there is a definable expression 𝑋 that also witnesses 𝐹 < 𝐺. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))
Theorem	cantnflem1a 9638*	Lemma for cantnf 9646. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    ⇒   ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅))
Theorem	cantnflem1b 9639*	Lemma for cantnf 9646. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    ⇒   ⊢ ((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) → 𝑋 ⊆ (𝑂‘𝑢))
Theorem	cantnflem1c 9640*	Lemma for cantnf 9646. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) (Proof shortened by AV, 4-Apr-2020.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    ⇒   ⊢ ((((𝜑 ∧ (suc 𝑢 ∈ dom 𝑂 ∧ (◡𝑂‘𝑋) ⊆ 𝑢)) ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹‘𝑥) ≠ ∅ ∧ (𝑂‘𝑢) ∈ 𝑥)) → 𝑥 ∈ (𝐺 supp ∅))
Theorem	cantnflem1d 9641*	Lemma for cantnf 9646. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘(𝑥 ∈ 𝐵 ↦ if(𝑥 ⊆ 𝑋, (𝐹‘𝑥), ∅))) ∈ (𝐻‘suc (◡𝑂‘𝑋)))
Theorem	cantnflem1 9642*	Lemma for cantnf 9646. This part of the proof is showing uniqueness of the Cantor normal form. We already know that the relation 𝑇 is a strict order, but we haven't shown it is a well-order yet. But being a strict order is enough to show that two distinct 𝐹, 𝐺 are 𝑇 -related as 𝐹 < 𝐺 or 𝐺 < 𝐹, and WLOG assuming that 𝐹 < 𝐺, we show that CNF respects this order and maps these two to different ordinals. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 2-Jul-2019.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐹 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → 𝐹𝑇𝐺)    &   ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)}    &   ⊢ 𝑂 = OrdIso( E , (𝐺 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (𝑂‘𝑘)) ·o (𝐺‘(𝑂‘𝑘))) +o 𝑧)), ∅)    ⇒   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ ((𝐴 CNF 𝐵)‘𝐺))
Theorem	cantnflem2 9643*	Lemma for cantnf 9646. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    ⇒   ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
Theorem	cantnflem3 9644*	Lemma for cantnf 9646. Here we show existence of Cantor normal forms. Assuming (by transfinite induction) that every number less than 𝐶 has a normal form, we can use oeeu 8567 to factor 𝐶 into the form ((𝐴 ↑o 𝑋) ·o 𝑌) +o 𝑍 where 0 < 𝑌 < 𝐴 and 𝑍 < (𝐴 ↑o 𝑋) (and a fortiori 𝑋 < 𝐵). Then since 𝑍 < (𝐴 ↑o 𝑋) ≤ (𝐴 ↑o 𝑋) ·o 𝑌 ≤ 𝐶, 𝑍 has a normal form, and by appending the term (𝐴 ↑o 𝑋) ·o 𝑌 using cantnfp1 9634 we get a normal form for 𝐶. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    &   ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}    &   ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))    &   ⊢ 𝑌 = (1st ‘𝑃)    &   ⊢ 𝑍 = (2nd ‘𝑃)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑆)    &   ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐺) = 𝑍)    &   ⊢ 𝐹 = (𝑡 ∈ 𝐵 ↦ if(𝑡 = 𝑋, 𝑌, (𝐺‘𝑡)))    ⇒   ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))
Theorem	cantnflem4 9645*	Lemma for cantnf 9646. Complete the induction step of cantnflem3 9644. (Contributed by Mario Carneiro, 25-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))    &   ⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))    &   ⊢ (𝜑 → ∅ ∈ 𝐶)    &   ⊢ 𝑋 = ∪ ∩ {𝑐 ∈ On ∣ 𝐶 ∈ (𝐴 ↑o 𝑐)}    &   ⊢ 𝑃 = (℩𝑑∃𝑎 ∈ On ∃𝑏 ∈ (𝐴 ↑o 𝑋)(𝑑 = ⟨𝑎, 𝑏⟩ ∧ (((𝐴 ↑o 𝑋) ·o 𝑎) +o 𝑏) = 𝐶))    &   ⊢ 𝑌 = (1st ‘𝑃)    &   ⊢ 𝑍 = (2nd ‘𝑃)    ⇒   ⊢ (𝜑 → 𝐶 ∈ ran (𝐴 CNF 𝐵))
Theorem	cantnf 9646*	The Cantor Normal Form theorem. The function (𝐴 CNF 𝐵), which maps a finitely supported function from 𝐵 to 𝐴 to the sum ((𝐴 ↑o 𝑓(𝑎1)) ∘ 𝑎1) +o ((𝐴 ↑o 𝑓(𝑎2)) ∘ 𝑎2) +o ... over all indices 𝑎 < 𝐵 such that 𝑓(𝑎) is nonzero, is an order isomorphism from the ordering 𝑇 of finitely supported functions to the set (𝐴 ↑o 𝐵) under the natural order. Setting 𝐴 = ω and letting 𝐵 be arbitrarily large, the surjectivity of this function implies that every ordinal has a Cantor normal form (and injectivity, together with coherence cantnfres 9630, implies that such a representation is unique). (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)))
Theorem	oemapwe 9647*	The lexicographic order on a function space of ordinals gives a well-ordering with order type equal to the ordinal exponential. This provides an alternate definition of the ordinal exponential. (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵)))
Theorem	cantnffval2 9648*	An alternate definition of df-cnf 9615 which relies on cantnf 9646. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9617 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆))
Theorem	cantnff1o 9649	Simplify the isomorphism of cantnf 9646 to simple bijection. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ 𝑆 = dom (𝐴 CNF 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ On)    ⇒   ⊢ (𝜑 → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵))
Theorem	wemapwe 9650*	Construct lexicographic order on a function space based on a reverse well-ordering of the indices and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑧𝑅𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}    &   ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍}    &   ⊢ (𝜑 → 𝑅 We 𝐴)    &   ⊢ (𝜑 → 𝑆 We 𝐵)    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ 𝐹 = OrdIso(𝑅, 𝐴)    &   ⊢ 𝐺 = OrdIso(𝑆, 𝐵)    &   ⊢ 𝑍 = (𝐺‘∅)    ⇒   ⊢ (𝜑 → 𝑇 We 𝑈)
Theorem	oef1o 9651*	A bijection of the base sets induces a bijection on ordinal exponentials. (The assumption (𝐹‘∅) = ∅ can be discharged using fveqf1o 7277.) (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐶)    &   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐷)    &   ⊢ (𝜑 → 𝐴 ∈ (On ∖ 1o))    &   ⊢ (𝜑 → 𝐵 ∈ On)    &   ⊢ (𝜑 → 𝐶 ∈ On)    &   ⊢ (𝜑 → 𝐷 ∈ On)    &   ⊢ (𝜑 → (𝐹‘∅) = ∅)    &   ⊢ 𝐾 = (𝑦 ∈ {𝑥 ∈ (𝐴 ↑m 𝐵) ∣ 𝑥 finSupp ∅} ↦ (𝐹 ∘ (𝑦 ∘ ◡𝐺)))    &   ⊢ 𝐻 = (((𝐶 CNF 𝐷) ∘ 𝐾) ∘ ◡(𝐴 CNF 𝐵))    ⇒   ⊢ (𝜑 → 𝐻:(𝐴 ↑o 𝐵)–1-1-onto→(𝐶 ↑o 𝐷))
Theorem	cnfcomlem 9652*	Lemma for cnfcom 9653. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)    &   ⊢ (𝜑 → 𝑂 ∈ (ω ↑o (𝐺‘𝐼)))    &   ⊢ (𝜑 → (𝑇‘𝐼):(𝐻‘𝐼)–1-1-onto→𝑂)    ⇒   ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
Theorem	cnfcom 9653*	Any ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. Here we show that bijection explicitly. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ (𝜑 → 𝐼 ∈ dom 𝐺)    ⇒   ⊢ (𝜑 → (𝑇‘suc 𝐼):(𝐻‘suc 𝐼)–1-1-onto→((ω ↑o (𝐺‘𝐼)) ·o (𝐹‘(𝐺‘𝐼))))
Theorem	cnfcom2lem 9654*	Lemma for cnfcom2 9655. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ∅ ∈ 𝐵)    ⇒   ⊢ (𝜑 → dom 𝐺 = suc ∪ dom 𝐺)
Theorem	cnfcom2 9655*	Any nonzero ordinal 𝐵 is equinumerous to the leading term of its Cantor normal form. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 3-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ∅ ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑇‘dom 𝐺):𝐵–1-1-onto→((ω ↑o 𝑊) ·o (𝐹‘𝑊)))
Theorem	cnfcom3lem 9656*	Lemma for cnfcom3 9657. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ω ⊆ 𝐵)    ⇒   ⊢ (𝜑 → 𝑊 ∈ (On ∖ 1o))
Theorem	cnfcom3 9657*	Any infinite ordinal 𝐵 is equinumerous to a power of ω. (We are being careful here to show explicit bijections rather than simple equinumerosity because we want a uniform construction for cnfcom3c 9659.) (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 4-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ (𝜑 → 𝐴 ∈ On)    &   ⊢ (𝜑 → 𝐵 ∈ (ω ↑o 𝐴))    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝐵)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ (𝜑 → ω ⊆ 𝐵)    &   ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢))    &   ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))    &   ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))    ⇒   ⊢ (𝜑 → 𝑁:𝐵–1-1-onto→(ω ↑o 𝑊))
Theorem	cnfcom3clem 9658*	Lemma for cnfcom3c 9659. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by AV, 4-Jul-2019.)
⊢ 𝑆 = dom (ω CNF 𝐴)    &   ⊢ 𝐹 = (◡(ω CNF 𝐴)‘𝑏)    &   ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅))    &   ⊢ 𝐻 = seqω((𝑘 ∈ V, 𝑧 ∈ V ↦ (𝑀 +o 𝑧)), ∅)    &   ⊢ 𝑇 = seqω((𝑘 ∈ V, 𝑓 ∈ V ↦ 𝐾), ∅)    &   ⊢ 𝑀 = ((ω ↑o (𝐺‘𝑘)) ·o (𝐹‘(𝐺‘𝑘)))    &   ⊢ 𝐾 = ((𝑥 ∈ 𝑀 ↦ (dom 𝑓 +o 𝑥)) ∪ ◡(𝑥 ∈ dom 𝑓 ↦ (𝑀 +o 𝑥)))    &   ⊢ 𝑊 = (𝐺‘∪ dom 𝐺)    &   ⊢ 𝑋 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((𝐹‘𝑊) ·o 𝑣) +o 𝑢))    &   ⊢ 𝑌 = (𝑢 ∈ (𝐹‘𝑊), 𝑣 ∈ (ω ↑o 𝑊) ↦ (((ω ↑o 𝑊) ·o 𝑢) +o 𝑣))    &   ⊢ 𝑁 = ((𝑋 ∘ ◡𝑌) ∘ (𝑇‘dom 𝐺))    &   ⊢ 𝐿 = (𝑏 ∈ (ω ↑o 𝐴) ↦ 𝑁)    ⇒   ⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
Theorem	cnfcom3c 9659*	Wrap the construction of cnfcom3 9657 into an existential quantifier. For any ω ⊆ 𝑏, there is a bijection from 𝑏 to some power of ω. Furthermore, this bijection is canonical , which means that we can find a single function 𝑔 which will give such bijections for every 𝑏 less than some arbitrarily large bound 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ (𝐴 ∈ On → ∃𝑔∀𝑏 ∈ 𝐴 (ω ⊆ 𝑏 → ∃𝑤 ∈ (On ∖ 1o)(𝑔‘𝑏):𝑏–1-1-onto→(ω ↑o 𝑤)))
2.6.4  Transitive closure of a relation
Syntax	cttrcl 9660	Declare the syntax for the transitive closure of a class.
class t++𝑅
Definition	df-ttrcl 9661*	Define the transitive closure of a class. This is the smallest relation containing 𝑅 (or more precisely, the relation (𝑅 ↾ V) induced by 𝑅) and having the transitive property. Definition from [Levy] p. 59, who denotes it as 𝑅∗ and calls it the "ancestral" of 𝑅. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑚 ∈ 𝑛 (𝑓‘𝑚)𝑅(𝑓‘suc 𝑚))}
Theorem	ttrcleq 9662	Equality theorem for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (𝑅 = 𝑆 → t++𝑅 = t++𝑆)
Theorem	nfttrcld 9663	Bound variable hypothesis builder for transitive closure. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ (𝜑 → Ⅎ𝑥𝑅)    ⇒   ⊢ (𝜑 → Ⅎ𝑥t++𝑅)
Theorem	nfttrcl 9664	Bound variable hypothesis builder for transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ Ⅎ𝑥𝑅    ⇒   ⊢ Ⅎ𝑥t++𝑅
Theorem	relttrcl 9665	The transitive closure of a class is a relation. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ Rel t++𝑅
Theorem	brttrcl 9666*	Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 18-Aug-2024.)
⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘𝑛) = 𝐵) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
Theorem	brttrcl2 9667*	Characterization of elements of the transitive closure of a relation. (Contributed by Scott Fenton, 24-Aug-2024.)
⊢ (𝐴t++𝑅𝐵 ↔ ∃𝑛 ∈ ω ∃𝑓(𝑓 Fn suc suc 𝑛 ∧ ((𝑓‘∅) = 𝐴 ∧ (𝑓‘suc 𝑛) = 𝐵) ∧ ∀𝑎 ∈ suc 𝑛(𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)))
Theorem	ssttrcl 9668	If 𝑅 is a relation, then it is a subclass of its transitive closure. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (Rel 𝑅 → 𝑅 ⊆ t++𝑅)
Theorem	ttrcltr 9669	The transitive closure of a class is transitive. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (t++𝑅 ∘ t++𝑅) ⊆ t++𝑅
Theorem	ttrclresv 9670	The transitive closure of 𝑅 restricted to V is the same as the transitive closure of 𝑅 itself. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ t++(𝑅 ↾ V) = t++𝑅
Theorem	ttrclco 9671	Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ (t++𝑅 ∘ 𝑅) ⊆ t++𝑅
Theorem	cottrcl 9672	Composition law for the transitive closure of a relation. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ (𝑅 ∘ t++𝑅) ⊆ t++𝑅
Theorem	ttrclss 9673	If 𝑅 is a subclass of 𝑆 and 𝑆 is transitive, then the transitive closure of 𝑅 is a subclass of 𝑆. (Contributed by Scott Fenton, 20-Oct-2024.)
⊢ ((𝑅 ⊆ 𝑆 ∧ (𝑆 ∘ 𝑆) ⊆ 𝑆) → t++𝑅 ⊆ 𝑆)
Theorem	dmttrcl 9674	The domain of a transitive closure is the same as the domain of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ dom t++𝑅 = dom 𝑅
Theorem	rnttrcl 9675	The range of a transitive closure is the same as the range of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ ran t++𝑅 = ran 𝑅
Theorem	ttrclexg 9676	If 𝑅 is a set, then so is t++𝑅. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ (𝑅 ∈ 𝑉 → t++𝑅 ∈ V)
Theorem	dfttrcl2 9677*	When 𝑅 is a set and a relation, then its transitive closure can be defined by an intersection. (Contributed by Scott Fenton, 26-Oct-2024.)
⊢ ((𝑅 ∈ 𝑉 ∧ Rel 𝑅) → t++𝑅 = ∩ {𝑧 ∣ (𝑅 ⊆ 𝑧 ∧ (𝑧 ∘ 𝑧) ⊆ 𝑧)})
Theorem	ttrclselem1 9678*	Lemma for ttrclse 9680. Show that all finite ordinal function values of 𝐹 are subsets of 𝐴. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   ⊢ (𝑁 ∈ ω → (𝐹‘𝑁) ⊆ 𝐴)
Theorem	ttrclselem2 9679*	Lemma for ttrclse 9680. Show that a suc 𝑁 element long chain gives membership in the 𝑁-th predecessor class and vice-versa. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ 𝐹 = rec((𝑏 ∈ V ↦ ∪ 𝑤 ∈ 𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))    ⇒   ⊢ ((𝑁 ∈ ω ∧ 𝑅 Se 𝐴 ∧ 𝑋 ∈ 𝐴) → (∃𝑓(𝑓 Fn suc suc 𝑁 ∧ ((𝑓‘∅) = 𝑦 ∧ (𝑓‘suc 𝑁) = 𝑋) ∧ ∀𝑎 ∈ suc 𝑁(𝑓‘𝑎)(𝑅 ↾ 𝐴)(𝑓‘suc 𝑎)) ↔ 𝑦 ∈ (𝐹‘𝑁)))
Theorem	ttrclse 9680	If 𝑅 is set-like over 𝐴, then the transitive closure of the restriction of 𝑅 to 𝐴 is set-like over 𝐴. This theorem requires the axioms of infinity and replacement for its proof. (Contributed by Scott Fenton, 31-Oct-2024.)
⊢ (𝑅 Se 𝐴 → t++(𝑅 ↾ 𝐴) Se 𝐴)
2.6.5  Transitive closure
Theorem	trcl 9681*	For any set 𝐴, show the properties of its transitive closure 𝐶. Similar to Theorem 9.1 of [TakeutiZaring] p. 73 except that we show an explicit expression for the transitive closure rather than just its existence. See tz9.1 9682 for an abbreviated version showing existence. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐹 = (rec((𝑧 ∈ V ↦ (𝑧 ∪ ∪ 𝑧)), 𝐴) ↾ ω)    &   ⊢ 𝐶 = ∪ 𝑦 ∈ ω (𝐹‘𝑦)    ⇒   ⊢ (𝐴 ⊆ 𝐶 ∧ Tr 𝐶 ∧ ∀𝑥((𝐴 ⊆ 𝑥 ∧ Tr 𝑥) → 𝐶 ⊆ 𝑥))
Theorem	tz9.1 9682*	Every set has a transitive closure (the smallest transitive extension). Theorem 9.1 of [TakeutiZaring] p. 73. See trcl 9681 for an explicit expression for the transitive closure. Apparently open problems are whether this theorem can be proved without the Axiom of Infinity; if not, then whether it implies Infinity; and if not, what is the "property" that Infinity has that the other axioms don't have that is weaker than Infinity itself? (Added 22-Mar-2011) The following article seems to answer the first question, that it can't be proved without Infinity, in the affirmative: Mancini, Antonella and Zambella, Domenico (2001). "A note on recursive models of set theories." Notre Dame Journal of Formal Logic, 42(2):109-115. (Thanks to Scott Fenton.) (Contributed by NM, 15-Sep-2003.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
Theorem	tz9.1c 9683*	Alternate expression for the existence of transitive closures tz9.1 9682: the intersection of all transitive sets containing 𝐴 is a set. (Contributed by Mario Carneiro, 22-Mar-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V
Theorem	epfrs 9684*	The strong form of the Axiom of Regularity (no sethood requirement on 𝐴), with the axiom itself present as an antecedent. See also zfregs 9685. (Contributed by Mario Carneiro, 22-Mar-2013.)
⊢ (( E Fr 𝐴 ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	zfregs 9685*	The strong form of the Axiom of Regularity, which does not require that 𝐴 be a set. Axiom 6' of [TakeutiZaring] p. 21. See also epfrs 9684. (Contributed by NM, 17-Sep-2003.)
⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝑥 ∩ 𝐴) = ∅)
Theorem	zfregs2 9686*	Alternate strong form of the Axiom of Regularity. Not every element of a nonempty class contains some element of that class. (Contributed by Alan Sare, 24-Oct-2011.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
⊢ (𝐴 ≠ ∅ → ¬ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥))
Theorem	setind 9687*	Set (epsilon) induction. Theorem 5.22 of [TakeutiZaring] p. 21. (Contributed by NM, 17-Sep-2003.)
⊢ (∀𝑥(𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → 𝐴 = V)
Theorem	setind2 9688	Set (epsilon) induction, stated compactly. Given as a homework problem in 1992 by George Boolos (1940-1996). (Contributed by NM, 17-Sep-2003.)
⊢ (𝒫 𝐴 ⊆ 𝐴 → 𝐴 = V)
Syntax	ctc 9689	Extend class notation to include the transitive closure function.
class TC
Definition	df-tc 9690*	The transitive closure function. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ TC = (𝑥 ∈ V ↦ ∩ {𝑦 ∣ (𝑥 ⊆ 𝑦 ∧ Tr 𝑦)})
Theorem	tcvalg 9691*	Value of the transitive closure function. (The fact that this intersection exists is a non-trivial fact that depends on ax-inf 9591; see tz9.1 9682.) (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → (TC‘𝐴) = ∩ {𝑥 ∣ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥)})
Theorem	tcid 9692	Defining property of the transitive closure function: it contains its argument as a subset. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (TC‘𝐴))
Theorem	tctr 9693	Defining property of the transitive closure function: it is transitive. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ Tr (TC‘𝐴)
Theorem	tcmin 9694	Defining property of the transitive closure function: it is a subset of any transitive class containing 𝐴. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (𝐴 ∈ 𝑉 → ((𝐴 ⊆ 𝐵 ∧ Tr 𝐵) → (TC‘𝐴) ⊆ 𝐵))
Theorem	tc2 9695*	A variant of the definition of the transitive closure function, using instead the smallest transitive set containing 𝐴 as a member, gives almost the same set, except that 𝐴 itself must be added because it is not usually a member of (TC‘𝐴) (and it is never a member if 𝐴 is well-founded). (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ((TC‘𝐴) ∪ {𝐴}) = ∩ {𝑥 ∣ (𝐴 ∈ 𝑥 ∧ Tr 𝑥)}
Theorem	tcsni 9696	The transitive closure of a singleton. Proof suggested by Gérard Lang. (Contributed by Mario Carneiro, 4-Jun-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (TC‘{𝐴}) = ((TC‘𝐴) ∪ {𝐴})
Theorem	tcss 9697	The transitive closure function inherits the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ⊆ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Theorem	tcel 9698	The transitive closure function converts the element relation to the subset relation. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐵 ∈ 𝐴 → (TC‘𝐵) ⊆ (TC‘𝐴))
Theorem	tcidm 9699	The transitive closure function is idempotent. (Contributed by Mario Carneiro, 23-Jun-2013.)
⊢ (TC‘(TC‘𝐴)) = (TC‘𝐴)
Theorem	tc0 9700	The transitive closure of the empty set. (Contributed by Mario Carneiro, 4-Jun-2015.)
⊢ (TC‘∅) = ∅