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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 4441 |
. 2
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2 | pwuni 4194 |
. . 3
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3 | pwexg 4182 |
. . 3
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4 | ssexg 4144 |
. . 3
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5 | 2, 3, 4 | sylancr 414 |
. 2
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6 | 1, 5 | impbii 126 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rex 2461 df-v 2741 df-in 3137 df-ss 3144 df-pw 3579 df-uni 3812 |
This theorem is referenced by: pwexb 4476 elpwpwel 4477 tfrlemibex 6332 tfr1onlembex 6348 tfrcllembex 6361 ixpexgg 6724 ptex 12718 tgss2 13664 txbasex 13842 |
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