P. 92 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 9101-9200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	domnsym 9101	Theorem 22(i) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.)
⊢ (𝐴 ≼ 𝐵 → ¬ 𝐵 ≺ 𝐴)
Theorem	0domg 9102	Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5362, ax-un 7727. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)
Theorem	0domgOLD 9103	Obsolete version of 0domg 9102 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → ∅ ≼ 𝐴)
Theorem	dom0 9104	A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) Avoid ax-pow 5362, ax-un 7727. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅)
Theorem	dom0OLD 9105	Obsolete version of dom0 9104 as of 29-Nov-2024. (Contributed by NM, 22-Nov-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≼ ∅ ↔ 𝐴 = ∅)
Theorem	0sdomg 9106	A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 23-Mar-2006.) Avoid ax-pow 5362, ax-un 7727. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	0sdomgOLD 9107	Obsolete version of 0sdomg 9106 as of 29-Nov-2024. (Contributed by NM, 23-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ∈ 𝑉 → (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅))
Theorem	0dom 9108	Any set dominates the empty set. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ 𝐴 ∈ V    ⇒   ⊢ ∅ ≼ 𝐴
Theorem	0sdom 9109	A set strictly dominates the empty set iff it is not empty. (Contributed by NM, 29-Jul-2004.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅)
Theorem	sdom0 9110	The empty set does not strictly dominate any set. (Contributed by NM, 26-Oct-2003.) Avoid ax-pow 5362, ax-un 7727. (Revised by BTernaryTau, 29-Nov-2024.)
⊢ ¬ 𝐴 ≺ ∅
Theorem	sdom0OLD 9111	Obsolete version of sdom0 9110 as of 29-Nov-2024. (Contributed by NM, 26-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝐴 ≺ ∅
Theorem	sdomdomtr 9112	Transitivity of strict dominance and dominance. Theorem 22(iii) of [Suppes] p. 97. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≺ 𝐶)
Theorem	sdomentr 9113	Transitivity of strict dominance and equinumerosity. Exercise 11 of [Suppes] p. 98. (Contributed by NM, 26-Oct-2003.)
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≺ 𝐶)
Theorem	domsdomtr 9114	Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97. (Contributed by NM, 10-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶)
Theorem	ensdomtr 9115	Transitivity of equinumerosity and strict dominance. (Contributed by NM, 26-Oct-2003.) (Revised by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶)
Theorem	sdomirr 9116	Strict dominance is irreflexive. Theorem 21(i) of [Suppes] p. 97. (Contributed by NM, 4-Jun-1998.)
⊢ ¬ 𝐴 ≺ 𝐴
Theorem	sdomtr 9117	Strict dominance is transitive. Theorem 21(iii) of [Suppes] p. 97. (Contributed by NM, 9-Jun-1998.)
⊢ ((𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐶) → 𝐴 ≺ 𝐶)
Theorem	sdomn2lp 9118	Strict dominance has no 2-cycle loops. (Contributed by NM, 6-May-2008.)
⊢ ¬ (𝐴 ≺ 𝐵 ∧ 𝐵 ≺ 𝐴)
Theorem	enen1 9119	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≈ 𝐶 ↔ 𝐵 ≈ 𝐶))
Theorem	enen2 9120	Equality-like theorem for equinumerosity. (Contributed by NM, 18-Dec-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≈ 𝐴 ↔ 𝐶 ≈ 𝐵))
Theorem	domen1 9121	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≼ 𝐶 ↔ 𝐵 ≼ 𝐶))
Theorem	domen2 9122	Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≼ 𝐴 ↔ 𝐶 ≼ 𝐵))
Theorem	sdomen1 9123	Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ≺ 𝐶 ↔ 𝐵 ≺ 𝐶))
Theorem	sdomen2 9124	Equality-like theorem for equinumerosity and strict dominance. (Contributed by NM, 8-Nov-2003.)
⊢ (𝐴 ≈ 𝐵 → (𝐶 ≺ 𝐴 ↔ 𝐶 ≺ 𝐵))
Theorem	domtriord 9125	Dominance is trichotomous in the restricted case of ordinal numbers. (Contributed by Jeff Hankins, 24-Oct-2009.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≼ 𝐵 ↔ ¬ 𝐵 ≺ 𝐴))
Theorem	sdomel 9126	For ordinals, strict dominance implies membership. (Contributed by Mario Carneiro, 13-Jan-2013.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ≺ 𝐵 → 𝐴 ∈ 𝐵))
Theorem	sdomdif 9127	The difference of a set from a smaller set cannot be empty. (Contributed by Mario Carneiro, 5-Feb-2013.)
⊢ (𝐴 ≺ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅)
Theorem	onsdominel 9128	An ordinal with more elements of some type is larger. (Contributed by Stefan O'Rear, 2-Nov-2014.)
⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ (𝐴 ∩ 𝐶) ≺ (𝐵 ∩ 𝐶)) → 𝐴 ∈ 𝐵)
Theorem	domunsn 9129	Dominance over a set with one element added. (Contributed by Mario Carneiro, 18-May-2015.)
⊢ (𝐴 ≺ 𝐵 → (𝐴 ∪ {𝐶}) ≼ 𝐵)
Theorem	fodomr 9130*	There exists a mapping from a set onto any (nonempty) set that it dominates. (Contributed by NM, 23-Mar-2006.)
⊢ ((∅ ≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)
Theorem	pwdom 9131	Injection of sets implies injection on power sets. (Contributed by Mario Carneiro, 9-Apr-2015.)
⊢ (𝐴 ≼ 𝐵 → 𝒫 𝐴 ≼ 𝒫 𝐵)
Theorem	canth2 9132	Cantor's Theorem. No set is equinumerous to its power set. Specifically, any set has a cardinality (size) strictly less than the cardinality of its power set. For example, the cardinality of real numbers is the same as the cardinality of the power set of integers, so real numbers cannot be put into a one-to-one correspondence with integers. Theorem 23 of [Suppes] p. 97. For the function version, see canth 7364. This is Metamath 100 proof #63. (Contributed by NM, 7-Aug-1994.)
⊢ 𝐴 ∈ V    ⇒   ⊢ 𝐴 ≺ 𝒫 𝐴
Theorem	canth2g 9133	Cantor's theorem with the sethood requirement expressed as an antecedent. Theorem 23 of [Suppes] p. 97. (Contributed by NM, 7-Nov-2003.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ≺ 𝒫 𝐴)
Theorem	2pwuninel 9134	The power set of the power set of the union of a set does not belong to the set. This theorem provides a way of constructing a new set that doesn't belong to a given set. (Contributed by NM, 27-Jun-2008.)
⊢ ¬ 𝒫 𝒫 ∪ 𝐴 ∈ 𝐴
Theorem	2pwne 9135	No set equals the power set of its power set. (Contributed by NM, 17-Nov-2008.)
⊢ (𝐴 ∈ 𝑉 → 𝒫 𝒫 𝐴 ≠ 𝐴)
Theorem	disjen 9136	A stronger form of pwuninel 8262. We can use pwuninel 8262, 2pwuninel 9134 to create one or two sets disjoint from a given set 𝐴, but here we show that in fact such constructions exist for arbitrarily large disjoint extensions, which is to say that for any set 𝐵 we can construct a set 𝑥 that is equinumerous to it and disjoint from 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ (𝐵 × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (𝐵 × {𝒫 ∪ ran 𝐴}) ≈ 𝐵))
Theorem	disjenex 9137*	Existence version of disjen 9136. (Contributed by Mario Carneiro, 7-Feb-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥((𝐴 ∩ 𝑥) = ∅ ∧ 𝑥 ≈ 𝐵))
Theorem	domss2 9138	A corollary of disjenex 9137. If 𝐹 is an injection from 𝐴 to 𝐵 then 𝐺 is a right inverse of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ 𝐺 = ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))    ⇒   ⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺:𝐵–1-1-onto→ran 𝐺 ∧ 𝐴 ⊆ ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)))
Theorem	domssex2 9139*	A corollary of disjenex 9137. If 𝐹 is an injection from 𝐴 to 𝐵 then there is a right inverse 𝑔 of 𝐹 from 𝐵 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑔(𝑔:𝐵–1-1→V ∧ (𝑔 ∘ 𝐹) = ( I ↾ 𝐴)))
Theorem	domssex 9140*	Weakening of domssex2 9139 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ (𝐴 ≼ 𝐵 → ∃𝑥(𝐴 ⊆ 𝑥 ∧ 𝐵 ≈ 𝑥))
2.4.31  Equinumerosity (cont.)
Theorem	xpf1o 9141*	Construct a bijection on a Cartesian product given bijections on the factors. (Contributed by Mario Carneiro, 30-May-2015.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝑋):𝐴–1-1-onto→𝐵)    &   ⊢ (𝜑 → (𝑦 ∈ 𝐶 ↦ 𝑌):𝐶–1-1-onto→𝐷)    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨𝑋, 𝑌⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))
Theorem	xpen 9142	Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))
Theorem	mapen 9143	Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by NM, 16-Dec-2003.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑m 𝐶) ≈ (𝐵 ↑m 𝐷))
Theorem	mapdom1 9144	Order-preserving property of set exponentiation. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 27-Jul-2004.) (Revised by Mario Carneiro, 9-Mar-2013.)
⊢ (𝐴 ≼ 𝐵 → (𝐴 ↑m 𝐶) ≼ (𝐵 ↑m 𝐶))
Theorem	mapxpen 9145	Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96. (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝐴 ↑m 𝐵) ↑m 𝐶) ≈ (𝐴 ↑m (𝐵 × 𝐶)))
Theorem	xpmapenlem 9146*	Lemma for xpmapen 9147. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    &   ⊢ 𝐷 = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧)))    &   ⊢ 𝑅 = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧)))    &   ⊢ 𝑆 = (𝑧 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)    ⇒   ⊢ ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴 ↑m 𝐶) × (𝐵 ↑m 𝐶))
Theorem	xpmapen 9147	Equinumerosity law for set exponentiation of a Cartesian product. Exercise 4.47 of [Mendelson] p. 255. (Contributed by NM, 23-Feb-2004.) (Proof shortened by Mario Carneiro, 16-Nov-2014.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    &   ⊢ 𝐶 ∈ V    ⇒   ⊢ ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴 ↑m 𝐶) × (𝐵 ↑m 𝐶))
Theorem	mapunen 9148	Equinumerosity law for set exponentiation of a disjoint union. Exercise 4.45 of [Mendelson] p. 255. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 29-Apr-2015.)
⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐶 ↑m (𝐴 ∪ 𝐵)) ≈ ((𝐶 ↑m 𝐴) × (𝐶 ↑m 𝐵)))
Theorem	map2xp 9149	A cardinal power with exponent 2 is equivalent to a Cartesian product with itself. (Contributed by Mario Carneiro, 31-May-2015.) (Proof shortened by AV, 17-Jul-2022.)
⊢ (𝐴 ∈ 𝑉 → (𝐴 ↑m 2o) ≈ (𝐴 × 𝐴))
Theorem	mapdom2 9150	Order-preserving property of set exponentiation. Theorem 6L(d) of [Enderton] p. 149. (Contributed by NM, 23-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝐴 ≼ 𝐵 ∧ ¬ (𝐴 = ∅ ∧ 𝐶 = ∅)) → (𝐶 ↑m 𝐴) ≼ (𝐶 ↑m 𝐵))
Theorem	mapdom3 9151	Set exponentiation dominates the base. (Contributed by Mario Carneiro, 30-Apr-2015.) (Proof shortened by AV, 17-Jul-2022.)
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴 ↑m 𝐵))
Theorem	pwen 9152	If two sets are equinumerous, then their power sets are equinumerous. Proposition 10.15 of [TakeutiZaring] p. 87. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 9-Apr-2015.)
⊢ (𝐴 ≈ 𝐵 → 𝒫 𝐴 ≈ 𝒫 𝐵)
Theorem	ssenen 9153*	Equinumerosity of equinumerous subsets of a set. (Contributed by NM, 30-Sep-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ (𝐴 ≈ 𝐵 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝐶)} ≈ {𝑥 ∣ (𝑥 ⊆ 𝐵 ∧ 𝑥 ≈ 𝐶)})
Theorem	limenpsi 9154	A limit ordinal is equinumerous to a proper subset of itself. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 16-Nov-2014.)
⊢ Lim 𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ (𝐴 ∖ {∅}))
Theorem	limensuci 9155	A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
⊢ Lim 𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴)
Theorem	limensuc 9156	A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.)
⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ≈ suc 𝐴)
Theorem	infensuc 9157	Any infinite ordinal is equinumerous to its successor. Exercise 7 of [TakeutiZaring] p. 88. Proved without the Axiom of Infinity. (Contributed by NM, 30-Oct-2003.) (Revised by Mario Carneiro, 13-Jan-2013.)
⊢ ((𝐴 ∈ On ∧ ω ⊆ 𝐴) → 𝐴 ≈ suc 𝐴)
2.4.32  Finite sets
Theorem	dif1enlem 9158	Lemma for rexdif1en 9160 and dif1en 9162. (Contributed by BTernaryTau, 18-Aug-2024.) Generalize to all ordinals and add a sethood requirement to avoid ax-un 7727. (Revised by BTernaryTau, 5-Jan-2025.)
⊢ (((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝑀 ∈ On) ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
Theorem	dif1enlemOLD 9159	Obsolete version of dif1enlem 9158 as of 5-Jan-2025. (Contributed by BTernaryTau, 18-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝑀 ∈ ω ∧ 𝐹:𝐴–1-1-onto→suc 𝑀) → (𝐴 ∖ {(◡𝐹‘𝑀)}) ≈ 𝑀)
Theorem	rexdif1en 9160*	If a set is equinumerous to a nonzero ordinal, then there exists an element in that set such that removing it leaves the set equinumerous to the predecessor of that ordinal. (Contributed by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals and avoid ax-un 7727. (Revised by BTernaryTau, 5-Jan-2025.)
⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
Theorem	rexdif1enOLD 9161*	Obsolete version of rexdif1en 9160 as of 5-Jan-2025. (Contributed by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ 𝐴 (𝐴 ∖ {𝑥}) ≈ 𝑀)
Theorem	dif1en 9162	If a set 𝐴 is equinumerous to the successor of an ordinal 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5362. (Revised by BTernaryTau, 26-Aug-2024.) Generalize to all ordinals. (Revised by BTernaryTau, 6-Jan-2025.)
⊢ ((𝑀 ∈ On ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Theorem	dif1ennn 9163	If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. See also dif1ennnALT 9279. (Contributed by BTernaryTau, 6-Jan-2025.)
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Theorem	dif1enOLD 9164	Obsolete version of dif1en 9162 as of 6-Jan-2025. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.) Avoid ax-pow 5362. (Revised by BTernaryTau, 26-Aug-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀 ∧ 𝑋 ∈ 𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Theorem	findcard 9165*	Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = (𝑦 ∖ {𝑧}) → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ Fin → (∀𝑧 ∈ 𝑦 𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ Fin → 𝜏)
Theorem	findcard2 9166*	Schema for induction on the cardinality of a finite set. The inductive step shows that the result is true if one more element is added to the set. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 8-Jul-2010.) Avoid ax-pow 5362. (Revised by BTernaryTau, 26-Aug-2024.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ (𝑦 ∈ Fin → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ Fin → 𝜏)
Theorem	findcard2s 9167*	Variation of findcard2 9166 requiring that the element added in the induction step not be a member of the original set. (Contributed by Paul Chapman, 30-Nov-2012.)
⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓))    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))    &   ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜑 ↔ 𝜃))    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏))    &   ⊢ 𝜓    &   ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜒 → 𝜃))    ⇒   ⊢ (𝐴 ∈ Fin → 𝜏)
Theorem	findcard2d 9168*	Deduction version of findcard2 9166. (Contributed by SO, 16-Jul-2018.)
⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))    &   ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))    &   ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))    &   ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))    &   ⊢ (𝜑 → 𝜒)    &   ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝜂)
Theorem	nnfi 9169	Natural numbers are finite sets. (Contributed by Stefan O'Rear, 21-Mar-2015.) Avoid ax-pow 5362. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ∈ ω → 𝐴 ∈ Fin)
Theorem	pssnn 9170*	A proper subset of a natural number is equinumerous to some smaller number. Lemma 6F of [Enderton] p. 137. (Contributed by NM, 22-Jun-1998.) (Revised by Mario Carneiro, 16-Nov-2014.) Avoid ax-pow 5362. (Revised by BTernaryTau, 31-Jul-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴) → ∃𝑥 ∈ 𝐴 𝐵 ≈ 𝑥)
Theorem	ssnnfi 9171	A subset of a natural number is finite. (Contributed by NM, 24-Jun-1998.) (Proof shortened by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
Theorem	ssnnfiOLD 9172	Obsolete version of ssnnfi 9171 as of 23-Sep-2024. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ ω ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
Theorem	0fin 9173	The empty set is finite. (Contributed by FL, 14-Jul-2008.)
⊢ ∅ ∈ Fin
Theorem	unfi 9174	The union of two finite sets is finite. Part of Corollary 6K of [Enderton] p. 144. (Contributed by NM, 16-Nov-2002.) Avoid ax-pow 5362. (Revised by BTernaryTau, 7-Aug-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
Theorem	ssfi 9175	A subset of a finite set is finite. Corollary 6G of [Enderton] p. 138. For a shorter proof using ax-pow 5362, see ssfiALT 9176. (Contributed by NM, 24-Jun-1998.) Avoid ax-pow 5362. (Revised by BTernaryTau, 12-Aug-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
Theorem	ssfiALT 9176	Shorter proof of ssfi 9175 using ax-pow 5362. (Contributed by NM, 24-Jun-1998.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin)
Theorem	imafi 9177	Images of finite sets are finite. For a shorter proof using ax-pow 5362, see imafiALT 9347. (Contributed by Stefan O'Rear, 22-Feb-2015.) Avoid ax-pow 5362. (Revised by BTernaryTau, 7-Sep-2024.)
⊢ ((Fun 𝐹 ∧ 𝑋 ∈ Fin) → (𝐹 “ 𝑋) ∈ Fin)
Theorem	pwfir 9178	If the power set of a set is finite, then the set itself is finite. (Contributed by BTernaryTau, 7-Sep-2024.)
⊢ (𝒫 𝐵 ∈ Fin → 𝐵 ∈ Fin)
Theorem	pwfilem 9179*	Lemma for pwfi 9180. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5362. (Revised by BTernaryTau, 7-Sep-2024.)
⊢ 𝐹 = (𝑐 ∈ 𝒫 𝑏 ↦ (𝑐 ∪ {𝑥}))    ⇒   ⊢ (𝒫 𝑏 ∈ Fin → 𝒫 (𝑏 ∪ {𝑥}) ∈ Fin)
Theorem	pwfi 9180	The power set of a finite set is finite and vice-versa. Theorem 38 of [Suppes] p. 104 and its converse, Theorem 40 of [Suppes] p. 105. (Contributed by NM, 26-Mar-2007.) Avoid ax-pow 5362. (Revised by BTernaryTau, 7-Sep-2024.)
⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
Theorem	diffi 9181	If 𝐴 is finite, (𝐴 ∖ 𝐵) is finite. (Contributed by FL, 3-Aug-2009.)
⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐵) ∈ Fin)
Theorem	cnvfi 9182	If a set is finite, its converse is as well. (Contributed by Mario Carneiro, 28-Dec-2014.) Avoid ax-pow 5362. (Revised by BTernaryTau, 9-Sep-2024.)
⊢ (𝐴 ∈ Fin → ◡𝐴 ∈ Fin)
Theorem	fnfi 9183	A version of fnex 7220 for finite sets that does not require Replacement or Power Sets. (Contributed by Mario Carneiro, 16-Nov-2014.) (Revised by Mario Carneiro, 24-Jun-2015.)
⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin)
Theorem	f1oenfi 9184	If the domain of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1oeng 8969). (Contributed by BTernaryTau, 8-Sep-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1oenfirn 9185	If the range of a one-to-one, onto function is finite, then the domain and range of the function are equinumerous. (Contributed by BTernaryTau, 9-Sep-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1-onto→𝐵) → 𝐴 ≈ 𝐵)
Theorem	f1domfi 9186	If the codomain of a one-to-one function is finite, then the function's domain is dominated by its codomain. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1domg 8970). (Contributed by BTernaryTau, 25-Sep-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
Theorem	f1domfi2 9187	If the domain of a one-to-one function is finite, then the function's domain is dominated by its codomain when the latter is a set. This theorem is proved without using the Axiom of Power Sets (unlike f1dom2g 8967). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ≼ 𝐵)
Theorem	enreffi 9188	Equinumerosity is reflexive for finite sets, proved without using the Axiom of Power Sets (unlike enrefg 8982). (Contributed by BTernaryTau, 8-Sep-2024.)
⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
Theorem	ensymfib 9189	Symmetry of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike ensymb 9000). (Contributed by BTernaryTau, 9-Sep-2024.)
⊢ (𝐴 ∈ Fin → (𝐴 ≈ 𝐵 ↔ 𝐵 ≈ 𝐴))
Theorem	entrfil 9190	Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 9004). (Contributed by BTernaryTau, 10-Sep-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	enfii 9191	A set equinumerous to a finite set is finite. (Contributed by Mario Carneiro, 12-Mar-2015.) Avoid ax-pow 5362. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵) → 𝐴 ∈ Fin)
Theorem	enfi 9192	Equinumerous sets have the same finiteness. For a shorter proof using ax-pow 5362, see enfiALT 9193. (Contributed by NM, 22-Aug-2008.) Avoid ax-pow 5362. (Revised by BTernaryTau, 23-Sep-2024.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
Theorem	enfiALT 9193	Shorter proof of enfi 9192 using ax-pow 5362. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
Theorem	domfi 9194	A set dominated by a finite set is finite. (Contributed by NM, 23-Mar-2006.) (Revised by Mario Carneiro, 12-Mar-2015.)
⊢ ((𝐴 ∈ Fin ∧ 𝐵 ≼ 𝐴) → 𝐵 ∈ Fin)
Theorem	entrfi 9195	Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 9004). (Contributed by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	entrfir 9196	Transitivity of equinumerosity for finite sets, proved without using the Axiom of Power Sets (unlike entr 9004). (Contributed by BTernaryTau, 23-Sep-2024.)
⊢ ((𝐶 ∈ Fin ∧ 𝐴 ≈ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≈ 𝐶)
Theorem	domtrfil 9197	Transitivity of dominance relation when 𝐴 is finite, proved without using the Axiom of Power Sets (unlike domtr 9005). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐴 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	domtrfi 9198	Transitivity of dominance relation when 𝐵 is finite, proved without using the Axiom of Power Sets (unlike domtr 9005). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐵 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	domtrfir 9199	Transitivity of dominance relation for finite sets, proved without using the Axiom of Power Sets (unlike domtr 9005). (Contributed by BTernaryTau, 24-Nov-2024.)
⊢ ((𝐶 ∈ Fin ∧ 𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)
Theorem	f1imaenfi 9200	If a function is one-to-one, then the image of a finite subset of its domain under it is equinumerous to the subset. This theorem is proved without using the Axiom of Replacement or the Axiom of Power Sets (unlike f1imaeng 9012). (Contributed by BTernaryTau, 29-Sep-2024.)
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴 ∧ 𝐶 ∈ Fin) → (𝐹 “ 𝐶) ≈ 𝐶)