Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > uniex2 | Structured version Visualization version Unicode version |
Description: The Axiom of Union using the standard abbreviation for union. Given any set , its union exists. (Contributed by NM, 4-Jun-2006.) |
Ref | Expression |
---|---|
uniex2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfun 6950 | . . . 4 | |
2 | eluni 4439 | . . . . . . 7 | |
3 | 2 | imbi1i 339 | . . . . . 6 |
4 | 3 | albii 1747 | . . . . 5 |
5 | 4 | exbii 1774 | . . . 4 |
6 | 1, 5 | mpbir 221 | . . 3 |
7 | 6 | bm1.3ii 4784 | . 2 |
8 | dfcleq 2616 | . . 3 | |
9 | 8 | exbii 1774 | . 2 |
10 | 7, 9 | mpbir 221 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 cuni 4436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-uni 4437 |
This theorem is referenced by: uniex 6953 |
Copyright terms: Public domain | W3C validator |