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Mirrors > Home > MPE Home > Th. List > inxp | Structured version Visualization version Unicode version |
Description: The intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
inxp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inopab 4983 |
. . 3
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2 | an4 838 |
. . . . 5
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3 | elin 3628 |
. . . . . 6
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4 | elin 3628 |
. . . . . 6
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5 | 3, 4 | anbi12i 708 |
. . . . 5
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6 | 2, 5 | bitr4i 260 |
. . . 4
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7 | 6 | opabbii 4480 |
. . 3
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8 | 1, 7 | eqtri 2483 |
. 2
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9 | df-xp 4858 |
. . 3
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10 | df-xp 4858 |
. . 3
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11 | 9, 10 | ineq12i 3643 |
. 2
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12 | df-xp 4858 |
. 2
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13 | 8, 11, 12 | 3eqtr4i 2493 |
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 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-9 1906 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 ax-sep 4538 ax-nul 4547 ax-pr 4652 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ne 2634 df-ral 2753 df-rex 2754 df-rab 2757 df-v 3058 df-dif 3418 df-un 3420 df-in 3422 df-ss 3429 df-nul 3743 df-if 3893 df-sn 3980 df-pr 3982 df-op 3986 df-opab 4475 df-xp 4858 df-rel 4859 |
This theorem is referenced by: xpindi 4986 xpindir 4987 dmxpin 5073 xpssres 5157 xpdisj1 5276 xpdisj2 5277 imainrect 5296 xpima 5297 curry1 6914 curry2 6917 fpar 6926 marypha1lem 7972 fpwwe2lem13 9092 hashxplem 12637 sscres 15776 gsumxp 17656 pjfval 19317 pjpm 19319 txbas 20630 txcls 20667 txrest 20694 trust 21292 ressuss 21326 trcfilu 21357 metreslem 21425 ressxms 21588 ressms 21589 mbfmcst 29129 0rrv 29332 poimirlem26 32010 xpheOLD 36421 |
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