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Mirrors > Home > ILE Home > Th. List > uniop | Unicode version |
Description: The union of an ordered pair. Theorem 65 of [Suppes] p. 39. (Contributed by NM, 17-Aug-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opthw.1 |
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opthw.2 |
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Ref | Expression |
---|---|
uniop |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opthw.1 |
. . . 4
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2 | opthw.2 |
. . . 4
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3 | 1, 2 | dfop 3668 |
. . 3
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4 | 3 | unieqi 3710 |
. 2
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5 | 1 | snex 4067 |
. . 3
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6 | prexg 4091 |
. . . 4
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7 | 1, 2, 6 | mp2an 420 |
. . 3
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8 | 5, 7 | unipr 3714 |
. 2
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9 | snsspr1 3632 |
. . 3
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10 | ssequn1 3210 |
. . 3
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11 | 9, 10 | mpbi 144 |
. 2
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12 | 4, 8, 11 | 3eqtri 2137 |
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 681 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-10 1464 ax-11 1465 ax-i12 1466 ax-bndl 1467 ax-4 1468 ax-14 1473 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 ax-sep 4004 ax-pow 4056 ax-pr 4089 |
This theorem depends on definitions: df-bi 116 df-3an 945 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-rex 2394 df-v 2657 df-un 3039 df-in 3041 df-ss 3048 df-pw 3476 df-sn 3497 df-pr 3498 df-op 3500 df-uni 3701 |
This theorem is referenced by: uniopel 4136 elvvuni 4561 dmrnssfld 4758 |
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