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| Mirrors > Home > MPE Home > Th. List > trv | Structured version Visualization version GIF version | ||
| Description: The universe is transitive. (Contributed by NM, 14-Sep-2003.) | 
| Ref | Expression | 
|---|---|
| trv | ⊢ Tr V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ssv 4008 | . 2 ⊢ ∪ V ⊆ V | |
| 2 | df-tr 5260 | . 2 ⊢ (Tr V ↔ ∪ V ⊆ V) | |
| 3 | 1, 2 | mpbir 231 | 1 ⊢ Tr V | 
| Colors of variables: wff setvar class | 
| Syntax hints: Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 Tr wtr 5259 | 
| 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-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-tr 5260 | 
| This theorem is referenced by: (None) | 
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