Theorem nregmodelaxext 45295

Description: 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.)

Hypotheses

Ref	Expression
nregmodel.1	⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
nregmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )

Assertion

Ref	Expression
nregmodelaxext	⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)

Distinct variable groups:   𝑥,𝑦,𝑧   𝑧,𝐹

Allowed substitution hints:   𝑅(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦)

Proof of Theorem nregmodelaxext

Step	Hyp	Ref	Expression
1		nregmodel.1	. . 3 ⊢ 𝐹 = (( I ↾ (V ∖ {∅, {∅}})) ∪ {⟨∅, {∅}⟩, ⟨{∅}, ∅⟩})
2	1	nregmodelf1o 45292	. 2 ⊢ 𝐹:V–1-1-onto→V
3		nregmodel.2	. 2 ⊢ 𝑅 = (◡𝐹 ∘ E )
4	2, 3	permaxext 45282	1 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  Vcvv 3441   ∖ cdif 3899   ∪ cun 3900  ∅c0 4286  {csn 4581  {cpr 4583  ⟨cop 4587   class class class wbr 5099   I cid 5519   E cep 5524  ^◡ccnv 5624   ↾ cres 5627   ∘ ccom 5629

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 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501

This theorem is referenced by: (None)