Theorem permaxpr 44984

Description: The Axiom of Pairing ax-pr 5371 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 6835	. 2 ⊢ (◡𝐹‘{𝑥, 𝑦}) ∈ V
2		breq2 5096	. . . 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 1920	. 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 3440	. . . . 5 ⊢ 𝑤 ∈ V
8		prex 5376	. . . . 5 ⊢ {𝑥, 𝑦} ∈ V
9	5, 6, 7, 8	brpermmodelcnv 44978	. . . 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 1795	. 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 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3436  {cpr 4579   class class class wbr 5092   E cep 5518  ^◡ccnv 5618   ∘ ccom 5623  –1-1-onto→wf1o 6481  ‘cfv 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-id 5514  df-eprel 5519  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490

This theorem is referenced by: (None)