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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 4265 |
. 2
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2 | pwuni 4027 |
. . 3
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3 | pwexg 4015 |
. . 3
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4 | ssexg 3978 |
. . 3
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5 | 2, 3, 4 | sylancr 405 |
. 2
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6 | 1, 5 | impbii 124 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
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 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-un 4260 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rex 2365 df-v 2621 df-in 3005 df-ss 3012 df-pw 3431 df-uni 3654 |
This theorem is referenced by: pwexb 4296 elpwpwel 4297 tfrlemibex 6094 tfr1onlembex 6110 tfrcllembex 6123 |
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