Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > xpexr2 | Structured version Visualization version Unicode version |
Description: If a nonempty Cartesian product is a set, so are both of its components. (Contributed by NM, 27-Aug-2006.) |
Ref | Expression |
---|---|
xpexr2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpnz 5553 | . 2 | |
2 | dmxp 5344 | . . . . . 6 | |
3 | 2 | adantl 482 | . . . . 5 |
4 | dmexg 7097 | . . . . . 6 | |
5 | 4 | adantr 481 | . . . . 5 |
6 | 3, 5 | eqeltrrd 2702 | . . . 4 |
7 | rnxp 5564 | . . . . . 6 | |
8 | 7 | adantl 482 | . . . . 5 |
9 | rnexg 7098 | . . . . . 6 | |
10 | 9 | adantr 481 | . . . . 5 |
11 | 8, 10 | eqeltrrd 2702 | . . . 4 |
12 | 6, 11 | anim12dan 882 | . . 3 |
13 | 12 | ancom2s 844 | . 2 |
14 | 1, 13 | sylan2br 493 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 c0 3915 cxp 5112 cdm 5114 crn 5115 |
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-8 1992 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 ax-un 6949 |
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-uni 4437 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: xpfir 8182 bj-xpnzex 32946 |
Copyright terms: Public domain | W3C validator |