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Mirrors > Home > MPE Home > Th. List > dmv | Structured version Visualization version GIF version |
Description: The domain of the universe is the universe. (Contributed by NM, 8-Aug-2003.) |
Ref | Expression |
---|---|
dmv | ⊢ dom V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3658 | . 2 ⊢ dom V ⊆ V | |
2 | dmi 5372 | . . 3 ⊢ dom I = V | |
3 | ssv 3658 | . . . 4 ⊢ I ⊆ V | |
4 | dmss 5355 | . . . 4 ⊢ ( I ⊆ V → dom I ⊆ dom V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ dom I ⊆ dom V |
6 | 2, 5 | eqsstr3i 3669 | . 2 ⊢ V ⊆ dom V |
7 | 1, 6 | eqssi 3652 | 1 ⊢ dom V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 Vcvv 3231 ⊆ wss 3607 I cid 5052 dom cdm 5143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-rel 5150 df-dm 5153 |
This theorem is referenced by: (None) |
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