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Mirrors > Home > MPE Home > Th. List > xpcan2 | Structured version Visualization version Unicode version |
Description: Cancellation law for Cartesian product. (Contributed by NM, 30-Aug-2011.) |
Ref | Expression |
---|---|
xpcan2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xp11 5569 | . . 3 | |
2 | eqid 2622 | . . . 4 | |
3 | 2 | biantru 526 | . . 3 |
4 | 1, 3 | syl6bbr 278 | . 2 |
5 | nne 2798 | . . 3 | |
6 | simpl 473 | . . . . 5 | |
7 | xpeq1 5128 | . . . . . . . . . 10 | |
8 | 0xp 5199 | . . . . . . . . . 10 | |
9 | 7, 8 | syl6eq 2672 | . . . . . . . . 9 |
10 | 9 | eqeq1d 2624 | . . . . . . . 8 |
11 | eqcom 2629 | . . . . . . . 8 | |
12 | 10, 11 | syl6bb 276 | . . . . . . 7 |
13 | 12 | adantr 481 | . . . . . 6 |
14 | df-ne 2795 | . . . . . . . 8 | |
15 | xpeq0 5554 | . . . . . . . . 9 | |
16 | orel2 398 | . . . . . . . . 9 | |
17 | 15, 16 | syl5bi 232 | . . . . . . . 8 |
18 | 14, 17 | sylbi 207 | . . . . . . 7 |
19 | 18 | adantl 482 | . . . . . 6 |
20 | 13, 19 | sylbid 230 | . . . . 5 |
21 | eqtr3 2643 | . . . . 5 | |
22 | 6, 20, 21 | syl6an 568 | . . . 4 |
23 | xpeq1 5128 | . . . 4 | |
24 | 22, 23 | impbid1 215 | . . 3 |
25 | 5, 24 | sylanb 489 | . 2 |
26 | 4, 25 | pm2.61ian 831 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wceq 1483 wne 2794 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-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 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-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: (None) |
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