Theorem wunpr 10682

Description: A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)
wunpr.3	⊢ (𝜑 → 𝐵 ∈ 𝑈)

Assertion

Ref	Expression
wunpr	⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Proof of Theorem wunpr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		wunpr.3	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
3		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
4		iswun 10677	. . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
5	4	ibi 270	. . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
6	5	simp3d 1160	. . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
7		simp3 1154	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)
8	7	ralimi 3102	. . 3 ⊢ (∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)
9	3, 6, 8	3syl 19	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)
10		preq1 4695	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
11	10	eleq1d 2850	. . 3 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝑦} ∈ 𝑈))
12		preq2 4696	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
13	12	eleq1d 2850	. . 3 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
14	11, 13	rspc2va 3596	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
15	1, 2, 9, 14	syl21anc 850	1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ∅c0 4288  𝒫 cpw 4558  {cpr 4587  ∪ cuni 4868  Tr wtr 5212  WUnicwun 10673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-v 3459  df-un 3912  df-ss 3924  df-sn 4586  df-pr 4588  df-uni 4869  df-tr 5213  df-wun 10675

This theorem is referenced by:  wunun  10683  wuntp  10684  wunsn  10689  wunop  10695  intwun  10708  wuncval2  10720