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Mirrors > Home > MPE Home > Th. List > xpdisj1 | Structured version Visualization version Unicode version |
Description: Cartesian products with disjoint sets are disjoint. (Contributed by NM, 13-Sep-2004.) |
Ref | Expression |
---|---|
xpdisj1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq1 5128 | . 2 | |
2 | inxp 5254 | . 2 | |
3 | 0xp 5199 | . . 3 | |
4 | 3 | eqcomi 2631 | . 2 |
5 | 1, 2, 4 | 3eqtr4g 2681 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 cin 3573 c0 3915 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-rab 2921 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 df-rel 5121 |
This theorem is referenced by: djudisj 5561 xpdisjres 29411 esum2dlem 30154 nosupbnd2lem1 31861 noetalem2 31864 noetalem3 31865 bj-2upln1upl 33012 disjxp1 39238 |
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