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Mirrors > Home > MPE Home > Th. List > elxp | Structured version Visualization version Unicode version |
Description: Membership in a Cartesian product. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
elxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 5120 | . . 3 | |
2 | 1 | eleq2i 2693 | . 2 |
3 | elopab 4983 | . 2 | |
4 | 2, 3 | bitri 264 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 cop 4183 copab 4712 cxp 5112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-opab 4713 df-xp 5120 |
This theorem is referenced by: elxp2 5132 elxp2OLD 5133 0nelxp 5143 0nelxpOLD 5144 0nelelxp 5145 rabxp 5154 elxp3 5169 elvv 5177 elvvv 5178 0xp 5199 dfres3 5403 xpdifid 5562 dfco2a 5635 elsnxp 5677 elsnxpOLD 5678 tpres 6466 elxp4 7110 elxp5 7111 opabex3d 7145 opabex3 7146 xp1st 7198 xp2nd 7199 poxp 7289 soxp 7290 xpsnen 8044 xpcomco 8050 xpassen 8054 dfac5lem1 8946 dfac5lem4 8949 axdc4lem 9277 fsum2dlem 14501 fprod2dlem 14710 numclwlk1lem2fo 27228 elima4 31679 brcart 32039 brimg 32044 dibelval3 36436 |
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