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Mirrors > Home > MPE Home > Th. List > wuntr | Structured version Visualization version GIF version |
Description: A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wuntr | ⊢ (𝑈 ∈ WUni → Tr 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iswun 9526 | . . 3 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))) | |
2 | 1 | ibi 256 | . 2 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))) |
3 | 2 | simp1d 1073 | 1 ⊢ (𝑈 ∈ WUni → Tr 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∅c0 3915 𝒫 cpw 4158 {cpr 4179 ∪ cuni 4436 Tr wtr 4752 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-in 3581 df-ss 3588 df-uni 4437 df-tr 4753 df-wun 9524 |
This theorem is referenced by: wunelss 9530 intwun 9557 |
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