Theorem prclaxpr 45974

Description: A class that is closed under the pairing operation models the Axiom of Pairing ax-pr 5391. 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 3455	. . . . . 6 ⊢ 𝑤 ∈ V
2	1	elpr 4609	. . . . 5 ⊢ (𝑤 ∈ {𝑥, 𝑦} ↔ (𝑤 = 𝑥 ∨ 𝑤 = 𝑦))
3	2	biimpri 231	. . . 4 ⊢ ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})
4	3	rgenw 3081	. . 3 ⊢ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})
5		eleq2 2850	. . . . . 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 3186	. . . 4 ⊢ (𝑧 = {𝑥, 𝑦} → (∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) ↔ ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ {𝑥, 𝑦})))
8	7	rspcev 3577	. . 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 704	. 2 ⊢ ({𝑥, 𝑦} ∈ 𝑀 → ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))
10	9	2ralimi 3133	1 ⊢ (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 {𝑥, 𝑦} ∈ 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∃𝑧 ∈ 𝑀 ∀𝑤 ∈ 𝑀 ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  {cpr 4586

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-v 3453  df-un 3904  df-sn 4585  df-pr 4587

This theorem is used by:  wfaxpr  45987