Theorem prclaxpr 44978

Description: A class that is closed under the pairing operation models the Axiom of Pairing ax-pr 5430. 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 3483	. . . . . 6 ⊢ 𝑤 ∈ V
2	1	elpr 4648	. . . . 5 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
3	2	biimpri 228	. . . 4 ⊢ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})
4	3	rgenw 3064	. . 3 ⊢ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})
5		eleq2 2829	. . . . . 6 ⊢ (𝑧 = {𝑥, 𝑦} → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ {𝑥, 𝑦}))
6	5	imbi2d 340	. . . . 5 ⊢ (𝑧 = {𝑥, 𝑦} → (((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})))
7	6	ralbidv 3177	. . . 4 ⊢ (𝑧 = {𝑥, 𝑦} → (∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})))
8	7	rspcev 3621	. . 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 691	. 2 ⊢ ({𝑥, 𝑦} ∈ 𝑀 → ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
10	9	2ralimi 3122	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 848   = wceq 1540   ∈ wcel 2108  ∀wral 3060  ∃wrex 3069  {cpr 4626

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-v 3481  df-un 3955  df-sn 4625  df-pr 4627

This theorem is referenced by:  wfaxpr  44991