Theorem wunpr 10112

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 10107	. . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
5	4	ibi 269	. . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
6	5	simp3d 1140	. . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
7		simp3 1134	. . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)
8	7	ralimi 3155	. . 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 18	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)
10		preq1 4650	. . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
11	10	eleq1d 2895	. . 3 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝑦} ∈ 𝑈))
12		preq2 4651	. . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
13	12	eleq1d 2895	. . 3 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
14	11, 13	rspc2va 3621	. 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
15	1, 2, 9, 14	syl21anc 835	1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3011  ∀wral 3133  ∅c0 4274  𝒫 cpw 4520  {cpr 4550  ∪ cuni 4819  Tr wtr 5153  WUnicwun 10103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3012  df-ral 3138  df-v 3483  df-un 3924  df-in 3926  df-ss 3935  df-sn 4549  df-pr 4551  df-uni 4820  df-tr 5154  df-wun 10105

This theorem is referenced by:  wunun  10113  wuntp  10114  wunsn  10119  wunop  10125  intwun  10138  wuncval2  10150