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Mirrors > Home > ILE Home > Th. List > uniexb | Unicode version |
Description: The Axiom of Union and its converse. A class is a set iff its union is a set. (Contributed by NM, 11-Nov-2003.) |
Ref | Expression |
---|---|
uniexb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 4193 | . 2 | |
2 | pwuni 3963 | . . 3 | |
3 | pwexg 3954 | . . 3 | |
4 | ssexg 3917 | . . 3 | |
5 | 2, 3, 4 | sylancr 405 | . 2 |
6 | 1, 5 | impbii 124 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 103 wcel 1433 cvv 2601 wss 2973 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-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-un 4188 |
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-uni 3602 |
This theorem is referenced by: pwexb 4224 tfrlemibex 5966 |
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