| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > wununi | Structured version Visualization version GIF version | ||
| Description: A weak universe is closed under union. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| Ref | Expression |
|---|---|
| wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
| wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| Ref | Expression |
|---|---|
| wununi | ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | unieq 4918 | . . 3 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
| 2 | 1 | eleq1d 2826 | . 2 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝑈 ↔ ∪ 𝐴 ∈ 𝑈)) |
| 3 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
| 4 | iswun 10744 | . . . . 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 1145 | . . 3 ⊢ (𝑈 ∈ WUni → ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)) |
| 7 | simp1 1137 | . . . 4 ⊢ ((∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∪ 𝑥 ∈ 𝑈) | |
| 8 | 7 | ralimi 3083 | . . 3 ⊢ (∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥 ∈ 𝑈 ∪ 𝑥 ∈ 𝑈) |
| 9 | 3, 6, 8 | 3syl 18 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 ∪ 𝑥 ∈ 𝑈) |
| 10 | wununi.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 11 | 2, 9, 10 | rspcdva 3623 | 1 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∅c0 4333 𝒫 cpw 4600 {cpr 4628 ∪ cuni 4907 Tr wtr 5259 WUnicwun 10740 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-v 3482 df-ss 3968 df-uni 4908 df-tr 5260 df-wun 10742 |
| This theorem is referenced by: wunun 10750 wunint 10755 wundm 10768 wunrn 10769 wunfv 10772 intwun 10775 wuncval2 10787 wunstr 17225 wunfunc 17946 wunnat 18004 catcoppccl 18162 catcfuccl 18163 catcxpccl 18252 |
| Copyright terms: Public domain | W3C validator |