Theorem prclaxpr 45620

Description: A class that is closed under the pairing operation models the Axiom of Pairing ax-pr 5405. Lemma II.2.4(4) of [Kunen2] p. 111. (Contributed by Eric Schmidt, 29-Sep-2025.)

Assertion

Ref	Expression
prclaxpr	⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 {𝑥, 𝑦} ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))

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

Allowed substitution hints:   𝑀(𝑥,𝑦,𝑤)

Proof of Theorem prclaxpr

Step	Hyp	Ref	Expression
1		vex 3467	. . . . . 6 ⊢ 𝑤 ∈ V
2	1	elpr 4619	. . . . 5 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
3	2	biimpri 231	. . . 4 ⊢ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})
4	3	rgenw 3089	. . 3 ⊢ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})
5		eleq2 2858	. . . . . 6 ⊢ (𝑧 = {𝑥, 𝑦} → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
6	5	imbi2d 343	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} → (((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})))
7	6	ralbidv 3194	. . . 4 ⊢ (𝑧 = {𝑥, 𝑦} → (∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})))
8	7	rspcev 3590	. . 3 ⊢ (({𝑥, 𝑦} ∈ 𝑀 ∧ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})) → ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
9	4, 8	mpan2 703	. 2 ⊢ ({𝑥, 𝑦} ∈ 𝑀 → ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
10	9	2ralimi 3141	1 ⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 {𝑥, 𝑦} ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))

Syntax hints:   → wi 4   ∨ wo 860   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {cpr 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-v 3465  df-un 3918  df-sn 4595  df-pr 4597

This theorem is referenced by:  wfaxpr  45633