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Mirrors > Home > ILE Home > Th. List > xpundi | Unicode version |
Description: Distributive law for cross product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.) |
Ref | Expression |
---|---|
xpundi |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 4503 |
. 2
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2 | df-xp 4503 |
. . . 4
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3 | df-xp 4503 |
. . . 4
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4 | 2, 3 | uneq12i 3192 |
. . 3
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5 | elun 3181 |
. . . . . . 7
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6 | 5 | anbi2i 450 |
. . . . . 6
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7 | andi 790 |
. . . . . 6
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8 | 6, 7 | bitri 183 |
. . . . 5
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9 | 8 | opabbii 3953 |
. . . 4
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10 | unopab 3965 |
. . . 4
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11 | 9, 10 | eqtr4i 2136 |
. . 3
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12 | 4, 11 | eqtr4i 2136 |
. 2
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13 | 1, 12 | eqtr4i 2136 |
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-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-tru 1315 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-nfc 2242 df-v 2657 df-un 3039 df-opab 3948 df-xp 4503 |
This theorem is referenced by: xpun 4558 djuassen 7017 xpdjuen 7018 |
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