Theorem permaxpr 45167

Description: The Axiom of Pairing ax-pr 5374 holds in permutation models. Part of Exercise II.9.2 of [Kunen2] p. 148. (Contributed by Eric Schmidt, 6-Nov-2025.)

Hypotheses

Ref	Expression
permmodel.1	⊢ 𝐹:V–1-1-onto→V
permmodel.2	⊢ 𝑅 = (◡𝐹 ∘ E )

Assertion

Ref	Expression
permaxpr	⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅𝑧)

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

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

Proof of Theorem permaxpr

Step	Hyp	Ref	Expression
1		fvex 6844	. 2 ⊢ (◡𝐹‘{𝑥, 𝑦}) ∈ V
2		breq2 5099	. . . 4 ⊢ (𝑧 = (◡𝐹‘{𝑥, 𝑦}) → (𝑤𝑅𝑧 ↔ 𝑤𝑅(◡𝐹‘{𝑥, 𝑦})))
3	2	imbi2d 340	. . 3 ⊢ (𝑧 = (◡𝐹‘{𝑥, 𝑦}) → (((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅𝑧) ↔ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅(◡𝐹‘{𝑥, 𝑦}))))
4	3	albidv 1921	. 2 ⊢ (𝑧 = (◡𝐹‘{𝑥, 𝑦}) → (∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅𝑧) ↔ ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅(◡𝐹‘{𝑥, 𝑦}))))
5		permmodel.1	. . . . 5 ⊢ 𝐹:V–1-1-onto→V
6		permmodel.2	. . . . 5 ⊢ 𝑅 = (◡𝐹 ∘ E )
7		vex 3441	. . . . 5 ⊢ 𝑤 ∈ V
8		prex 5379	. . . . 5 ⊢ {𝑥, 𝑦} ∈ V
9	5, 6, 7, 8	brpermmodelcnv 45161	. . . 4 ⊢ (𝑤𝑅(◡𝐹‘{𝑥, 𝑦}) ↔ 𝑤 ∈ {𝑥, 𝑦})
10	7	elpr 4602	. . . 4 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
11	9, 10	sylbbr 236	. . 3 ⊢ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅(◡𝐹‘{𝑥, 𝑦}))
12	11	ax-gen 1796	. 2 ⊢ ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅(◡𝐹‘{𝑥, 𝑦}))
13	1, 4, 12	ceqsexv2d 3488	1 ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤𝑅𝑧)

Syntax hints:   → wi 4   ∨ wo 847  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3437  {cpr 4579   class class class wbr 5095   E cep 5520  ^◡ccnv 5620   ∘ ccom 5625  –1-1-onto→wf1o 6488  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-eprel 5521  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497

This theorem is referenced by: (None)