Theorem nregmodelaxext 45697

Description: The Axiom of Extensionality ax-ext 2733 is true in the permutation model defined from 𝐹. This theorem is an immediate consequence of the fact that ax-ext 2733 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 45694	. 2 ⊢ 𝐹:V–1-1-onto→V
3		nregmodel.2	. 2 ⊢ 𝑅 = (◡𝐹 ∘ E )
4	2, 3	permaxext 45684	1 ⊢ (∀𝑧(𝑧𝑅𝑥 ↔ 𝑧𝑅𝑦) → 𝑥 = 𝑦)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1566   = wceq 1568  Vcvv 3453   ∖ cdif 3901   ∪ cun 3902  ∅c0 4285  {csn 4588  {cpr 4590  ⟨cop 4594   class class class wbr 5108   I cid 5555   E cep 5560  ^◡ccnv 5660   ↾ cres 5663   ∘ ccom 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is referenced by: (None)