P. 454 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 45301-45400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	relprel 45301	A relation-preserving function preserves the relation. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 → (𝐻‘𝐶)𝑆(𝐻‘𝐷)))
Theorem	relpmin 45302	A preimage of a minimal element under a relation-preserving function is minimal. Essentially one half of isomin 7293. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ ((𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (((𝐻 “ 𝐶) ∩ (◡𝑆 “ {(𝐻‘𝐷)})) = ∅ → (𝐶 ∩ (◡𝑅 “ {𝐷})) = ∅))
Theorem	relpfrlem 45303*	Lemma for relpfr 45304. Proved without using the Axiom of Replacement. This is isofrlem 7296 with weaker hypotheses. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝜑 → 𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵))    &   ⊢ (𝜑 → (𝐻 “ 𝑥) ∈ V)    ⇒   ⊢ (𝜑 → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
Theorem	relpfr 45304	If the image of a set under a relation-preserving function is well-founded, so is the set. See isofr 7298 for a bidirectional statement. A more general version of Lemma I.9.9 of [Kunen2] p. 47. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ (𝐻 RelPres 𝑅, 𝑆(𝐴, 𝐵) → (𝑆 Fr 𝐵 → 𝑅 Fr 𝐴))
21.42.4  Orbits
Theorem	orbitex 45305	Orbits exist. Given a set 𝐴 and a function 𝐹, the orbit of 𝐴 under 𝐹 is the smallest set 𝑍 such that 𝐴 ∈ 𝑍 and 𝑍 is closed under 𝐹. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ (rec(𝐹, 𝐴) “ ω) ∈ V
Theorem	orbitinit 45306	A set is contained in its orbit. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ (rec(𝐹, 𝐴) “ ω))
Theorem	orbitcl 45307	The orbit under a function is closed under the function. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ (𝐵 ∈ (rec(𝐹, 𝐴) “ ω) → (𝐹‘𝐵) ∈ (rec(𝐹, 𝐴) “ ω))
Theorem	orbitclmpt 45308	Version of orbitcl 45307 using maps-to notation. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐷    &   ⊢ 𝑍 = (rec((𝑥 ∈ V ↦ 𝐶), 𝐴) “ ω)    &   ⊢ (𝑥 = 𝐵 → 𝐶 = 𝐷)    ⇒   ⊢ ((𝐵 ∈ 𝑍 ∧ 𝐷 ∈ 𝑉) → 𝐷 ∈ 𝑍)
21.42.5  Well-founded sets
Theorem	trwf 45309	The class of well-founded sets is transitive. (Contributed by Eric Schmidt, 9-Sep-2025.)
⊢ Tr ∪ (𝑅1 “ On)
Theorem	rankrelp 45310	The rank function preserves ∈. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ rank RelPres E , E (∪ (𝑅1 “ On), On)
Theorem	wffr 45311	The class of well-founded sets is well-founded. Lemma I.9.24(2) of [Kunen2] p. 53. (Contributed by Eric Schmidt, 11-Oct-2025.)
⊢ E Fr ∪ (𝑅1 “ On)
Theorem	trfr 45312	A transitive class well-founded by ∈ is a subclass of the class of well-founded sets. Part of Lemma I.9.21 of [Kunen2] p. 53. (Contributed by Eric Schmidt, 26-Oct-2025.)
⊢ ((Tr 𝐴 ∧ E Fr 𝐴) → 𝐴 ⊆ ∪ (𝑅1 “ On))
Theorem	tcfr 45313	A set is well-founded if and only if its transitive closure is well-founded by ∈. This characterization of well-founded sets is that in Definition I.9.20 of [Kunen2] p. 53. (Contributed by Eric Schmidt, 26-Oct-2025.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ E Fr (TC‘𝐴))
Theorem	xpwf 45314	The Cartesian product of two well-founded sets is well-founded. (Contributed by Eric Schmidt, 12-Sep-2025.)
⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ ∪ (𝑅1 “ On)) → (𝐴 × 𝐵) ∈ ∪ (𝑅1 “ On))
Theorem	dmwf 45315	The domain of a well-founded set is well-founded. (Contributed by Eric Schmidt, 12-Sep-2025.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → dom 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	rnwf 45316	The range of a well-founded set is well-founded. (Contributed by Eric Schmidt, 12-Sep-2025.)
⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → ran 𝐴 ∈ ∪ (𝑅1 “ On))
Theorem	relwf 45317	A relation is a well-founded set iff its domain and range are. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (Rel 𝑅 → (𝑅 ∈ ∪ (𝑅1 “ On) ↔ (dom 𝑅 ∈ ∪ (𝑅1 “ On) ∧ ran 𝑅 ∈ ∪ (𝑅1 “ On))))
21.42.6  Absoluteness in transitive models
Theorem	ralabso 45318*	Simplification of restricted quantification in a transitive class. When 𝜑 is quantifier-free, this shows that the formula ∀𝑥 ∈ 𝑦𝜑 is absolute for transitive models, which is a particular case of Lemma I.16.2 of [Kunen2] p. 95. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜑)))
Theorem	rexabso 45319*	Simplification of restricted quantification in a transitive class. When 𝜑 is quantifier-free, this shows that the formula ∃𝑥 ∈ 𝑦𝜑 is absolute for transitive models, which is a particular case of Lemma I.16.2 of [Kunen2] p. 95. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜑)))
Theorem	ralabsod 45320*	Deduction form of ralabso 45318. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝜑 → Tr 𝑀)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜓)))
Theorem	rexabsod 45321*	Deduction form of rexabso 45319. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝜑 → Tr 𝑀)    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜓)))
Theorem	ralabsobidv 45322*	Formula-building lemma for proving absoluteness results. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝜑 → Tr 𝑀)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝜒)))
Theorem	rexabsobidv 45323*	Formula-building lemma for proving absoluteness results. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝜑 → Tr 𝑀)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑀) → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 ∧ 𝜒)))
Theorem	ssabso 45324*	The notion "𝑥 is a subset of 𝑦 " is absolute for transitive models. Compare Example I.16.3 of [Kunen2] p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)))
Theorem	disjabso 45325*	Disjointness is absolute for transitive models. Compare Example I.16.3 of [Kunen2] p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → ((𝐴 ∩ 𝐵) = ∅ ↔ ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝐴 → ¬ 𝑥 ∈ 𝐵)))
Theorem	n0abso 45326*	Nonemptiness is absolute for transitive models. Compare Example I.16.3 of [Kunen2] p. 96 and the following discussion. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ 𝐴 ∈ 𝑀) → (𝐴 ≠ ∅ ↔ ∃𝑥 ∈ 𝑀 𝑥 ∈ 𝐴))
21.42.7  Lemmas for showing axioms hold in models
Theorem	traxext 45327*	A transitive class models the Axiom of Extensionality ax-ext 2709. Lemma II.2.4(1) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 11-Sep-2025.)
⊢ (Tr 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
Theorem	modelaxreplem1 45328*	Lemma for modelaxrep 45331. We show that 𝑀 is closed under taking subsets. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜓 → 𝑥 ⊆ 𝑀)    &   ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))    &   ⊢ (𝜓 → ∅ ∈ 𝑀)    &   ⊢ (𝜓 → 𝑥 ∈ 𝑀)    &   ⊢ 𝐴 ⊆ 𝑥    ⇒   ⊢ (𝜓 → 𝐴 ∈ 𝑀)
Theorem	modelaxreplem2 45329*	Lemma for modelaxrep 45331. We define a class 𝐹 and show that the antecedent of Replacement implies that 𝐹 is a function. We use Replacement (in the form of funex 7175) to show that 𝐹 exists. Then we show that, under our hypotheses, the range of 𝐹 is a member of 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜓 → 𝑥 ⊆ 𝑀)    &   ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))    &   ⊢ (𝜓 → ∅ ∈ 𝑀)    &   ⊢ (𝜓 → 𝑥 ∈ 𝑀)    &   ⊢ Ⅎ𝑤𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ Ⅎ𝑧𝐹    &   ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}    &   ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))    ⇒   ⊢ (𝜓 → ran 𝐹 ∈ 𝑀)
Theorem	modelaxreplem3 45330*	Lemma for modelaxrep 45331. We show that the consequent of Replacement is satisfied with ran 𝐹 as the value of 𝑦. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜓 → 𝑥 ⊆ 𝑀)    &   ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))    &   ⊢ (𝜓 → ∅ ∈ 𝑀)    &   ⊢ (𝜓 → 𝑥 ∈ 𝑀)    &   ⊢ Ⅎ𝑤𝜓    &   ⊢ Ⅎ𝑧𝜓    &   ⊢ Ⅎ𝑧𝐹    &   ⊢ 𝐹 = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ 𝑥 ∧ (𝑧 ∈ 𝑀 ∧ ∀𝑦𝜑))}    &   ⊢ (𝜓 → (𝑤 ∈ 𝑀 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦)))    ⇒   ⊢ (𝜓 → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
Theorem	modelaxrep 45331*	Conditions which guarantee that a class models the Axiom of Replacement ax-rep 5226. Similar to Lemma II.2.4(6) of [Kunen2] p. 111. The first two hypotheses are those in Kunen. The reason for the third hypothesis that our version of Replacement is different from Kunen's (which is zfrep6 7909). If we assumed Regularity, we could eliminate this extra hypothesis, since under Regularity, the empty set is a member of every non-empty transitive class. Note that, to obtain the relativization of an instance of Replacement to 𝑀, the formula ∀𝑦𝜑 would need to be replaced with ∀𝑦 ∈ 𝑀𝜒, where 𝜒 is 𝜑 with all quantifiers relativized to 𝑀. However, we can obtain this by using 𝑦 ∈ 𝑀 ∧ 𝜒 for 𝜑 in this theorem, so it does establish that all instances of Replacement hold in 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (𝜓 → Tr 𝑀)    &   ⊢ (𝜓 → ∀𝑓((Fun 𝑓 ∧ dom 𝑓 ∈ 𝑀 ∧ ran 𝑓 ⊆ 𝑀) → ran 𝑓 ∈ 𝑀))    &   ⊢ (𝜓 → ∅ ∈ 𝑀)    ⇒   ⊢ (𝜓 → ∀𝑥 ∈ 𝑀 (∀𝑤 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑀 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑))))
Theorem	ssclaxsep 45332*	A class that is closed under subsets models the Axiom of Separation ax-sep 5243. Lemma II.2.4(3) of [Kunen2] p. 111. Note that, to obtain the relativization of an instance of Separation to 𝑀, the formula 𝜑 would need to be replaced with its relativization to 𝑀. However, this new formula is a valid substitution for 𝜑, so this theorem does establish that all instances of Separation hold in 𝑀. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (∀𝑧 ∈ 𝑀 𝒫 𝑧 ⊆ 𝑀 → ∀𝑧 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑥 ∈ 𝑀 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑)))
Theorem	0elaxnul 45333*	A class that contains the empty set models the Null Set Axiom ax-nul 5253. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (∅ ∈ 𝑀 → ∃𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ¬ 𝑦 ∈ 𝑥)
Theorem	pwclaxpow 45334*	Suppose 𝑀 is a transitive class that is closed under power sets intersected with 𝑀. Then, 𝑀 models the Axiom of Power Sets ax-pow 5312. One direction of Lemma II.2.8 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ ∀𝑥 ∈ 𝑀 (𝒫 𝑥 ∩ 𝑀) ∈ 𝑀) → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
Theorem	prclaxpr 45335*	A class that is closed under the pairing operation models the Axiom of Pairing ax-pr 5379. Lemma II.2.4(4) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 {𝑥, 𝑦} ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
Theorem	uniclaxun 45336*	A class that is closed under the union operation models the Axiom of Union ax-un 7690. Lemma II.2.4(5) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 1-Oct-2025.)
⊢ (∀𝑥 ∈ 𝑀 ∪ 𝑥 ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (∃𝑤 ∈ 𝑀 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦))
Theorem	sswfaxreg 45337*	A subclass of the class of well-founded sets models the Axiom of Regularity ax-reg 9509. Lemma II.2.4(2) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (𝑀 ⊆ ∪ (𝑅1 “ On) → ∀𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 𝑦 ∈ 𝑥 → ∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥))))
Theorem	omssaxinf2 45338*	A class that contains all ordinals up to and including ω models the Axiom of Infinity ax-inf2 9562. The antecedent of this theorem is not enough to guarantee that the class models the alternate axiom ax-inf 9559. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((ω ⊆ 𝑀 ∧ ω ∈ 𝑀) → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))
Theorem	omelaxinf2 45339*	A transitive class that contains ω models the Axiom of Infinity ax-inf2 9562. Lemma II.2.11(7) of [Kunen2] p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in 𝑀. This, apparently, is because Kunen's statement of the Axiom of Infinity uses the defined notions ∅ and suc, and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears. The antecedent of this theorem is not enough to guarantee that the class models the alternate axiom ax-inf 9559. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((Tr 𝑀 ∧ ω ∈ 𝑀) → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))
Theorem	dfac5prim 45340*	dfac5 10051 expanded into primitives. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (CHOICE ↔ ∀𝑥((∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤 𝑤 ∈ 𝑧) ∧ ∀𝑧∀𝑤((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤∀𝑣((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))))
Theorem	ac8prim 45341*	ac8 10414 expanded into primitives. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ ((∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤 𝑤 ∈ 𝑧) ∧ ∀𝑧∀𝑤((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦∀𝑧(𝑧 ∈ 𝑥 → ∃𝑤∀𝑣((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))
Theorem	modelac8prim 45342*	If 𝑀 is a transitive class, then the following are equivalent. (1) Every nonempty set 𝑥 ∈ 𝑀 of pairwise disjoint nonempty sets has a choice set in 𝑀. (2) The class 𝑀 models the Axiom of Choice, in the form ac8prim 45341. Lemma II.2.11(7) of [Kunen2] p. 114. Kunen has the additional hypotheses that the Extensionality, Separation, Pairing, and Union axioms are true in 𝑀. This, apparently, is because Kunen's statement of the Axiom of Choice uses defined notions, including ∅ and ∩, and these axioms guarantee that these notions are well-defined. When we state the axiom using primitives only, the need for these hypotheses disappears. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ (Tr 𝑀 → (∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝑀 ((∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑀 ∀𝑣 ∈ 𝑀 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))))
21.42.8  The class of well-founded sets is a model for ZFC
Theorem	wfaxext 45343*	The class of well-founded sets models the Axiom of Extensionality ax-ext 2709. Part of Corollary II.2.5 of [Kunen2] p. 112. This is the first of a series of theorems showing that all the axioms of ZFC hold in the class of well-founded sets, which we here denote by 𝑊. More precisely, for each axiom of ZFC, we obtain a provable statement if we restrict all quantifiers to 𝑊 (including implicit universal quantifiers on free variables). None of these proofs use the Axiom of Regularity. In particular, the Axiom of Regularity itself is proved to hold in 𝑊 without using Regularity. Further, the Axiom of Choice is used only in the proof that Choice holds in 𝑊. This has the consequence that any theorem of ZF (possibly proved using Regularity) can be proved, without using Regularity, to hold in 𝑊. This gives us a relative consistency result: If ZF without Regularity is consistent, so is ZF itself. Similarly, if ZFC without Regularity is consistent, so is ZFC itself. These consistency results are metatheorems and are part of Theorem II.2.13 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 11-Sep-2025.) (Revised by Eric Schmidt, 29-Sep-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 (∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)
Theorem	wfaxrep 45344*	The class of well-founded sets models the Axiom of Replacement ax-rep 5226. Actually, our statement is stronger, since it is an instance of Replacement only when all quantifiers in ∀𝑦𝜑 are relativized to 𝑊. Essentially part of Corollary II.2.5 of [Kunen2] p. 112, but note that our Replacement is different from Kunen's. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 (∀𝑤 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 ↔ ∃𝑤 ∈ 𝑊 (𝑤 ∈ 𝑥 ∧ ∀𝑦𝜑)))
Theorem	wfaxsep 45345*	The class of well-founded sets models the Axiom of Separation ax-sep 5243. Actually, our statement is stronger, since it is an instance of Separation only when all quantifiers in 𝜑 are relativized to 𝑊. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑧 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑥 ∈ 𝑊 (𝑥 ∈ 𝑦 ↔ (𝑥 ∈ 𝑧 ∧ 𝜑))
Theorem	wfaxnul 45346*	The class of well-founded sets models the Null Set Axiom ax-nul 5253. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∃𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 ¬ 𝑦 ∈ 𝑥
Theorem	wfaxpow 45347*	The class of well-founded sets models the Axioms of Power Sets. Part of Corollary II.2.9 of [Kunen2] p. 113. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	wfaxpr 45348*	The class of well-founded sets models the Axiom of Pairing ax-pr 5379. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 29-Sep-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 ∃𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧)
Theorem	wfaxun 45349*	The class of well-founded sets models the Axiom of Union ax-un 7690. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (∃𝑤 ∈ 𝑊 (𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝑥) → 𝑧 ∈ 𝑦)
Theorem	wfaxreg 45350*	The class of well-founded sets models the Axiom of Regularity ax-reg 9509. Part of Corollary II.2.5 of [Kunen2] p. 112. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 (∃𝑦 ∈ 𝑊 𝑦 ∈ 𝑥 → ∃𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑦 → ¬ 𝑧 ∈ 𝑥)))
Theorem	wfaxinf2 45351*	The class of well-founded sets models the Axiom of Infinity ax-inf2 9562. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∃𝑥 ∈ 𝑊 (∃𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑊 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑊 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
Theorem	wfac8prim 45352*	The class of well-founded sets 𝑊 models the Axiom of Choice. Since the previous theorems show that all the ZF axioms hold in 𝑊, we may use any statement that ZF proves is equivalent to Choice to prove this. We use ac8prim 45341. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)
⊢ 𝑊 = ∪ (𝑅1 “ On)    ⇒   ⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 ∀𝑣 ∈ 𝑊 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))
21.42.9  Permutation models
Theorem	brpermmodel 45353	The membership relation in a permutation model. We use a permutation 𝐹 of the universe to define a relation 𝑅 that serves as the membership relation in our model. The conclusion of this theorem is Definition II.9.1 of [Kunen2] p. 148. All the axioms of ZFC except for Regularity hold in permutation models, and Regularity will be false if 𝐹 is chosen appropriately. Thus, permutation models can be used to show that Regularity does not follow from the other axioms (with the usual proviso that the axioms are consistent). (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴𝑅𝐵 ↔ 𝐴 ∈ (𝐹‘𝐵))
Theorem	brpermmodelcnv 45354	Ordinary membership expressed in terms of the permutation model's membership relation. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    &   ⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴𝑅(◡𝐹‘𝐵) ↔ 𝐴 ∈ 𝐵)
Theorem	permaxext 45355*	The Axiom of Extensionality ax-ext 2709 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)
Theorem	permaxrep 45356*	The Axiom of Replacement ax-rep 5226 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. Note that, to prove that an instance of Replacement holds in the model, 𝜑 would need have all instances of ∈ replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Replacement holds. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ (∀𝑤∃𝑦∀𝑧(∀𝑦𝜑 → 𝑧 = 𝑦) → ∃𝑦∀𝑧(𝑧𝑅𝑦 ↔ ∃𝑤(𝑤𝑅𝑥 ∧ ∀𝑦𝜑)))
Theorem	permaxsep 45357*	The Axiom of Separation ax-sep 5243 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. Note that, to prove that an instance of Separation holds in the model, 𝜑 would need have all instances of ∈ replaced with 𝑅. But this still results in an instance of this theorem, so we do establish that Separation holds. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑦∀𝑥(𝑥𝑅𝑦 ↔ (𝑥𝑅𝑧 ∧ 𝜑))
Theorem	permaxnul 45358*	The Null Set Axiom ax-nul 5253 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑥∀𝑦 ¬ 𝑦𝑅𝑥
Theorem	permaxpow 45359*	The Axiom of Power Sets ax-pow 5312 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑦∀𝑧(∀𝑤(𝑤𝑅𝑧 → 𝑤𝑅𝑥) → 𝑧𝑅𝑦)
Theorem	permaxpr 45360*	The Axiom of Pairing ax-pr 5379 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅𝑧)
Theorem	permaxun 45361*	The Axiom of Union ax-un 7690 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑦∀𝑧(∃𝑤(𝑧𝑅𝑤 ∧ 𝑤𝑅𝑥) → 𝑧𝑅𝑦)
Theorem	permaxinf2lem 45362*	Lemma for permaxinf2 45363. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    &   ⊢ 𝑍 = (rec((𝑣 ∈ V ↦ (◡𝐹‘((𝐹‘𝑣) ∪ {𝑣}))), (◡𝐹‘∅)) “ ω)    ⇒   ⊢ ∃𝑥(∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
Theorem	permaxinf2 45363*	The Axiom of Infinity ax-inf2 9562 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ∃𝑥(∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧 ¬ 𝑧𝑅𝑦) ∧ ∀𝑦(𝑦𝑅𝑥 → ∃𝑧(𝑧𝑅𝑥 ∧ ∀𝑤(𝑤𝑅𝑧 ↔ (𝑤𝑅𝑦 ∨ 𝑤 = 𝑦)))))
Theorem	permac8prim 45364*	The Axiom of Choice ac8prim 45341 holds in permutation models. Part of Exercise II.9.3 of [Kunen2] p. 149. Note that ax-ac 10381 requires Regularity for its derivation from the usual Axiom of Choice and does not necessarily hold in permutation models. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹:V–1-1-onto→V    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ((∀𝑧(𝑧𝑅𝑥 → ∃𝑤 𝑤𝑅𝑧) ∧ ∀𝑧∀𝑤((𝑧𝑅𝑥 ∧ 𝑤𝑅𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦(𝑦𝑅𝑧 → ¬ 𝑦𝑅𝑤)))) → ∃𝑦∀𝑧(𝑧𝑅𝑥 → ∃𝑤∀𝑣((𝑣𝑅𝑧 ∧ 𝑣𝑅𝑦) ↔ 𝑣 = 𝑤)))
Theorem	nregmodelf1o 45365	Define a permutation 𝐹 used to produce a model in which ax-reg 9509 is false. The permutation swaps ∅ and {∅} and leaves the rest of 𝑉 fixed. This is an example given after Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    ⇒   ⊢ 𝐹:V–1-1-onto→V
Theorem	nregmodellem 45366	Lemma for nregmodel 45367. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ (𝑥𝑅∅ ↔ 𝑥 ∈ {∅})
Theorem	nregmodel 45367*	The Axiom of Regularity ax-reg 9509 is false in the permutation model defined from 𝐹. Since the other axioms of ZFC hold in all permutation models (permaxext 45355 through permac8prim 45364), we can conclude that Regularity does not follow from those axioms, assuming ZFC is consistent. (If we could prove Regularity from the other axioms, we could prove it in the permutation model and thus obtain a contradiction with this theorem.) Since we also know that Regularity is consistent with the other axioms (wfaxext 45343 through wfac8prim 45352), Regularity is neither provable nor disprovable from the other axioms; i.e., it is independent of them. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ ¬ ∀𝑥(∃𝑦 𝑦𝑅𝑥 → ∃𝑦(𝑦𝑅𝑥 ∧ ∀𝑧(𝑧𝑅𝑦 → ¬ 𝑧𝑅𝑥)))
Theorem	nregmodelaxext 45368*	The Axiom of Extensionality ax-ext 2709 is true in the permutation model defined from 𝐹. This theorem is an immediate consequence of the fact that ax-ext 2709 holds in all permutation models and is provided as an illustration. (Contributed by Eric Schmidt, 16-Nov-2025.)
⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})    &   ⊢ 𝑅 = (◡𝐹 ∘ E )    ⇒   ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)
21.43  Mathbox for Glauco Siliprandi
21.43.1  Miscellanea
Theorem	evth2f 45369*	A version of evth2 24927 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑦𝐹    &   ⊢ Ⅎ𝑥𝑋    &   ⊢ Ⅎ𝑦𝑋    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑥) ≤ (𝐹‘𝑦))
Theorem	elunif 45370*	A version of eluni 4868 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 ∈ ∪ 𝐵 ↔ ∃𝑥(𝐴 ∈ 𝑥 ∧ 𝑥 ∈ 𝐵))
Theorem	rzalf 45371	A version of rzal 4449 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥 𝐴 = ∅    ⇒   ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)
Theorem	fvelrnbf 45372	A version of fvelrnb 6902 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐹    ⇒   ⊢ (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝐵))
Theorem	rfcnpre1 45373	If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than a given extended real B is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ 𝐵 < (𝐹‘𝑥)}    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐽)
Theorem	ubelsupr 45374*	If U belongs to A and U is an upper bound, then U is the sup of A. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ ((𝐴 ⊆ ℝ ∧ 𝑈 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝑥 ≤ 𝑈) → 𝑈 = sup(𝐴, ℝ, < ))
Theorem	fsumcnf 45375*	A finite sum of functions to complex numbers from a common topological space is continuous, without disjoint var constraint x ph. The class expression for B normally contains free variables k and x to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ 𝐾 = (TopOpen‘ℂfld)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
Theorem	mulltgt0 45376	The product of a negative and a positive number is negative. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ (((𝐴 ∈ ℝ ∧ 𝐴 < 0) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵)) → (𝐴 · 𝐵) < 0)
Theorem	rspcegf 45377	A version of rspcev 3578 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝜓    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    &   ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))    ⇒   ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)
Theorem	rabexgf 45378	A version of rabexg 5284 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝜑} ∈ V)
Theorem	fcnre 45379	A function continuous with respect to the standard topology, is a real mapping. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝑇 = ∪ 𝐽    &   ⊢ 𝐶 = (𝐽 Cn 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹:𝑇⟶ℝ)
Theorem	sumsnd 45380*	A sum of a singleton is the term. The deduction version of sumsn 15681. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ (𝜑 → Ⅎ𝑘𝐵)    &   ⊢ Ⅎ𝑘𝜑    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ {𝑀}𝐴 = 𝐵)
Theorem	evthf 45381*	A version of evth 24926 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑦𝐹    &   ⊢ Ⅎ𝑥𝑋    &   ⊢ Ⅎ𝑦𝑋    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑦𝜑    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑋 ≠ ∅)    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝐹‘𝑦) ≤ (𝐹‘𝑥))
Theorem	cnfex 45382	The class of continuous functions between two topologies is a set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top) → (𝐽 Cn 𝐾) ∈ V)
Theorem	fnchoice 45383*	For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ (𝐴 ∈ Fin → ∃𝑓(𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ≠ ∅ → (𝑓‘𝑥) ∈ 𝑥)))
Theorem	refsumcn 45384*	A finite sum of continuous real functions, from a common topological space, is continuous. The class expression for B normally contains free variables k and x to index it. See fsumcn 24829 for the analogous theorem on continuous complex functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ Σ𝑘 ∈ 𝐴 𝐵) ∈ (𝐽 Cn 𝐾))
Theorem	rfcnpre2 45385	If 𝐹 is a continuous function with respect to the standard topology, then the preimage A of the values smaller than a given extended real 𝐵, is an open set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐵    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝑋 = ∪ 𝐽    &   ⊢ 𝐴 = {𝑥 ∈ 𝑋 ∣ (𝐹‘𝑥) < 𝐵}    &   ⊢ (𝜑 → 𝐵 ∈ ℝ*)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐽)
Theorem	cncmpmax 45386*	When the hypothesis for the extreme value theorem hold, then the sup of the range of the function belongs to the range, it is real and it an upper bound of the range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ 𝑇 = ∪ 𝐽    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝑇 ≠ ∅)    ⇒   ⊢ (𝜑 → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ sup(ran 𝐹, ℝ, < ) ∈ ℝ ∧ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) ≤ sup(ran 𝐹, ℝ, < )))
Theorem	rfcnpre3 45387*	If F is a continuous function with respect to the standard topology, then the preimage A of the values greater than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑡𝐹    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝑇 = ∪ 𝐽    &   ⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ 𝐵 ≤ (𝐹‘𝑡)}    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
Theorem	rfcnpre4 45388*	If F is a continuous function with respect to the standard topology, then the preimage A of the values less than or equal to a given real B is a closed set. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑡𝐹    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ 𝑇 = ∪ 𝐽    &   ⊢ 𝐴 = {𝑡 ∈ 𝑇 ∣ (𝐹‘𝑡) ≤ 𝐵}    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
Theorem	sumpair 45389*	Sum of two distinct complex values. The class expression for 𝐴 and 𝐵 normally contain free variable 𝑘 to index it. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ (𝜑 → Ⅎ𝑘𝐷)    &   ⊢ (𝜑 → Ⅎ𝑘𝐸)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐷 ∈ ℂ)    &   ⊢ (𝜑 → 𝐸 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸)    ⇒   ⊢ (𝜑 → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸))
Theorem	rfcnnnub 45390*	Given a real continuous function 𝐹 defined on a compact topological space, there is always a positive integer that is a strict upper bound of its range. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑡𝐹    &   ⊢ Ⅎ𝑡𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ Comp)    &   ⊢ 𝑇 = ∪ 𝐽    &   ⊢ (𝜑 → 𝑇 ≠ ∅)    &   ⊢ 𝐶 = (𝐽 Cn 𝐾)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∀𝑡 ∈ 𝑇 (𝐹‘𝑡) < 𝑛)
Theorem	refsum2cnlem1 45391*	This is the core Lemma for refsum2cn 45392: the sum of two continuous real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐺    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐴 = (𝑘 ∈ {1, 2} ↦ if(𝑘 = 1, 𝐹, 𝐺))    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ (𝐽 Cn 𝐾))
Theorem	refsum2cn 45392*	The sum of two continuus real functions (from a common topological space) is continuous. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
⊢ Ⅎ𝑥𝐹    &   ⊢ Ⅎ𝑥𝐺    &   ⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐾 = (topGen‘ran (,))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ((𝐹‘𝑥) + (𝐺‘𝑥))) ∈ (𝐽 Cn 𝐾))
Theorem	adantlllr 45393	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((((𝜑 ∧ 𝜂) ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	3adantlr3 45394	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒)) ∧ 𝜃) → 𝜏)    ⇒   ⊢ (((𝜑 ∧ (𝜓 ∧ 𝜒 ∧ 𝜂)) ∧ 𝜃) → 𝜏)
Theorem	3adantll2 45395	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((((𝜑 ∧ 𝜂 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	3adantll3 45396	Deduction adding a conjunct to antecedent. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((((𝜑 ∧ 𝜓) ∧ 𝜒) ∧ 𝜃) → 𝜏)    ⇒   ⊢ ((((𝜑 ∧ 𝜓 ∧ 𝜂) ∧ 𝜒) ∧ 𝜃) → 𝜏)
Theorem	ssnel 45397	If not element of a set, then not element of a subset. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐶 ∈ 𝐵) → ¬ 𝐶 ∈ 𝐴)
Theorem	sncldre 45398	A singleton is closed w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ (𝐴 ∈ ℝ → {𝐴} ∈ (Clsd‘(topGen‘ran (,))))
Theorem	n0p 45399	A polynomial with a nonzero coefficient is not the zero polynomial. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ ((𝑃 ∈ (Poly‘ℤ) ∧ 𝑁 ∈ ℕ0 ∧ ((coeff‘𝑃)‘𝑁) ≠ 0) → 𝑃 ≠ 0𝑝)
Theorem	pm2.65ni 45400	Inference rule for proof by contradiction. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
⊢ (¬ 𝜑 → 𝜓)    &   ⊢ (¬ 𝜑 → ¬ 𝜓)    ⇒   ⊢ 𝜑