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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 4399 | . 2 | |
2 | pwuni 4153 | . . 3 | |
3 | pwexg 4141 | . . 3 | |
4 | ssexg 4103 | . . 3 | |
5 | 2, 3, 4 | sylancr 411 | . 2 |
6 | 1, 5 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wb 104 wcel 2128 cvv 2712 wss 3102 cpw 3543 cuni 3772 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rex 2441 df-v 2714 df-in 3108 df-ss 3115 df-pw 3545 df-uni 3773 |
This theorem is referenced by: pwexb 4434 elpwpwel 4435 tfrlemibex 6276 tfr1onlembex 6292 tfrcllembex 6305 ixpexgg 6667 tgss2 12479 txbasex 12657 |
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