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Mirrors > Home > MPE Home > Th. List > wunpr | Structured version Visualization version GIF version |
Description: A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
wunpr.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Ref | Expression |
---|---|
wunpr | ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
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 10742 | . . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
5 | 4 | ibi 267 | . . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
6 | 5 | simp3d 1143 | . . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
7 | simp3 1137 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) | |
8 | 7 | ralimi 3081 | . . 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 4738 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦}) | |
11 | 10 | eleq1d 2824 | . . 3 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝑦} ∈ 𝑈)) |
12 | preq2 4739 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵}) | |
13 | 12 | eleq1d 2824 | . . 3 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈)) |
14 | 11, 13 | rspc2va 3634 | . 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈) |
15 | 1, 2, 9, 14 | syl21anc 838 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∅c0 4339 𝒫 cpw 4605 {cpr 4633 ∪ cuni 4912 Tr wtr 5265 WUnicwun 10738 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-v 3480 df-un 3968 df-ss 3980 df-sn 4632 df-pr 4634 df-uni 4913 df-tr 5266 df-wun 10740 |
This theorem is referenced by: wunun 10748 wuntp 10749 wunsn 10754 wunop 10760 intwun 10773 wuncval2 10785 |
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