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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 4290 |
. 2
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2 | pwuni 4048 |
. . 3
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3 | pwexg 4036 |
. . 3
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4 | ssexg 3999 |
. . 3
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5 | 2, 3, 4 | sylancr 406 |
. 2
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6 | 1, 5 | impbii 125 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-un 4284 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-rex 2376 df-v 2635 df-in 3019 df-ss 3026 df-pw 3451 df-uni 3676 |
This theorem is referenced by: pwexb 4324 elpwpwel 4325 tfrlemibex 6132 tfr1onlembex 6148 tfrcllembex 6161 ixpexgg 6519 tgss2 11947 |
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