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Mirrors > Home > MPE Home > Th. List > univ | Structured version Visualization version 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 4709 | . . 3 ⊢ 𝒫 V = V | |
2 | 1 | unieqi 4721 | . 2 ⊢ ∪ 𝒫 V = ∪ V |
3 | unipw 5199 | . 2 ⊢ ∪ 𝒫 V = V | |
4 | 2, 3 | eqtr3i 2804 | 1 ⊢ ∪ V = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 Vcvv 3415 𝒫 cpw 4422 ∪ cuni 4712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-rex 3094 df-v 3417 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-pw 4424 df-sn 4442 df-pr 4444 df-uni 4713 |
This theorem is referenced by: (None) |
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