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| Mirrors > Home > MPE Home > Th. List > wunelss | Structured version Visualization version GIF version | ||
| Description: The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.) |
| Ref | Expression |
|---|---|
| wununi.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
| wununi.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
| Ref | Expression |
|---|---|
| wunelss | ⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | wununi.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
| 2 | wuntr 10745 | . . 3 ⊢ (𝑈 ∈ WUni → Tr 𝑈) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → Tr 𝑈) |
| 4 | wununi.2 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
| 5 | trss 5270 | . 2 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈)) | |
| 6 | 3, 4, 5 | sylc 65 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝑈) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ⊆ wss 3951 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: wunss 10752 wunf 10767 wuncval2 10787 |
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