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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 9405 | . . . . 5 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
5 | 4 | ibi 255 | . . . 4 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
6 | 5 | simp3d 1068 | . . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
7 | simp3 1056 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) | |
8 | 7 | ralimi 2936 | . . 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 4212 | . . . 4 ⊢ (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦}) | |
11 | 10 | eleq1d 2672 | . . 3 ⊢ (𝑥 = 𝐴 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝑦} ∈ 𝑈)) |
12 | preq2 4213 | . . . 4 ⊢ (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵}) | |
13 | 12 | eleq1d 2672 | . . 3 ⊢ (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈)) |
14 | 11, 13 | rspc2va 3294 | . 2 ⊢ (((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) ∧ ∀𝑥 ∈ 𝑈 ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈) |
15 | 1, 2, 9, 14 | syl21anc 1317 | 1 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∅c0 3874 𝒫 cpw 4108 {cpr 4127 ∪ cuni 4372 Tr wtr 4680 WUnicwun 9401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-v 3175 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 df-uni 4373 df-tr 4681 df-wun 9403 |
This theorem is referenced by: wunun 9411 wuntp 9412 wunsn 9417 wunop 9423 intwun 9436 wuncval2 9448 |
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