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Mirrors > Home > MPE Home > Th. List > wunun | Structured version Visualization version GIF version |
Description: A weak universe is closed under binary union. (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 |
---|---|
wunun | ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wununi.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | wunpr.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
3 | uniprg 4450 | . . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) | |
4 | 1, 2, 3 | syl2anc 693 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵)) |
5 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
6 | 5, 1, 2 | wunpr 9531 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈) |
7 | 5, 6 | wununi 9528 | . 2 ⊢ (𝜑 → ∪ {𝐴, 𝐵} ∈ 𝑈) |
8 | 4, 7 | eqeltrrd 2702 | 1 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 {cpr 4179 ∪ cuni 4436 WUnicwun 9522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-v 3202 df-un 3579 df-in 3581 df-ss 3588 df-sn 4178 df-pr 4180 df-uni 4437 df-tr 4753 df-wun 9524 |
This theorem is referenced by: wuntp 9533 wunsuc 9539 wunfi 9543 wunxp 9546 wuntpos 9556 wunsets 15900 catcoppccl 16758 |
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