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Mirrors > Home > MPE Home > Th. List > xpundir | Structured version Unicode version |
Description: Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.) |
Ref | Expression |
---|---|
xpundir |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 4944 |
. 2
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2 | df-xp 4944 |
. . . 4
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3 | df-xp 4944 |
. . . 4
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4 | 2, 3 | uneq12i 3606 |
. . 3
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5 | elun 3595 |
. . . . . . 7
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6 | 5 | anbi1i 695 |
. . . . . 6
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7 | andir 863 |
. . . . . 6
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8 | 6, 7 | bitri 249 |
. . . . 5
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9 | 8 | opabbii 4454 |
. . . 4
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10 | unopab 4465 |
. . . 4
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11 | 9, 10 | eqtr4i 2483 |
. . 3
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12 | 4, 11 | eqtr4i 2483 |
. 2
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13 | 1, 12 | eqtr4i 2483 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1952 ax-ext 2430 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2601 df-v 3070 df-un 3431 df-opab 4449 df-xp 4944 |
This theorem is referenced by: xpun 4994 resundi 5222 xpfi 7684 cdaassen 8452 hashxplem 12297 ustund 19912 cnmpt2pc 20616 pwssplit4 29580 xpprsng 30857 |
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