Theorem wfaxext 45271

Description: The class of well-founded sets models the Axiom of Extensionality ax-ext 2707. 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.)

Hypothesis

Ref	Expression
wfax.1	⊢ 𝑊 = ∪ (𝑅1 “ On)

Assertion

Ref	Expression
wfaxext	⊢ ∀𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 (∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Distinct variable group:   𝑥,𝑦,𝑧,𝑊

Proof of Theorem wfaxext

Step	Hyp	Ref	Expression
1		trwf 45237	. . 3 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . 4 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5211	. . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On)))
4	2, 3	ax-mp 5	. . 3 ⊢ (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On))
5	1, 4	mpbir 231	. 2 ⊢ Tr 𝑊
6		traxext 45255	. 2 ⊢ (Tr 𝑊 → ∀𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 (∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))
7	5, 6	ax-mp 5	1 ⊢ ∀𝑥 ∈ 𝑊 ∀𝑦 ∈ 𝑊 (∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542  ∀wral 3050  ∪ cuni 4862  Tr wtr 5204   “ cima 5626  Oncon0 6316  𝑅₁cr1 9676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-r1 9678

This theorem is referenced by: (None)