Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dmv | 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 3114 | . 2 ⊢ dom V ⊆ V | |
2 | dmi 4749 | . . 3 ⊢ dom I = V | |
3 | ssv 3114 | . . . 4 ⊢ I ⊆ V | |
4 | dmss 4733 | . . . 4 ⊢ ( I ⊆ V → dom I ⊆ dom V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ dom I ⊆ dom V |
6 | 2, 5 | eqsstrri 3125 | . 2 ⊢ V ⊆ dom V |
7 | 1, 6 | eqssi 3108 | 1 ⊢ dom V = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 Vcvv 2681 ⊆ wss 3066 I cid 4205 dom cdm 4534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-dm 4544 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |