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Mirrors > Home > ILE Home > Th. List > univ | GIF version |
Description: The union of the universe is the universe. Exercise 4.12(c) of [Mendelson] p. 235. (Contributed by NM, 14-Sep-2003.) |
Ref | Expression |
---|---|
univ | ⊢ ∪ V = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwv 3600 | . . 3 ⊢ 𝒫 V = V | |
2 | 1 | unieqi 3611 | . 2 ⊢ ∪ 𝒫 V = ∪ V |
3 | unipw 3972 | . 2 ⊢ ∪ 𝒫 V = V | |
4 | 2, 3 | eqtr3i 2103 | 1 ⊢ ∪ V = V |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 Vcvv 2601 𝒫 cpw 3382 ∪ cuni 3601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-uni 3602 |
This theorem is referenced by: (None) |
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