Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > xp0 | Structured version Visualization version GIF version |
Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
Ref | Expression |
---|---|
xp0 | ⊢ (𝐴 × ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xp 5122 | . . 3 ⊢ (∅ × 𝐴) = ∅ | |
2 | 1 | cnveqi 5219 | . 2 ⊢ ◡(∅ × 𝐴) = ◡∅ |
3 | cnvxp 5470 | . 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅) | |
4 | cnv0 5454 | . 2 ⊢ ◡∅ = ∅ | |
5 | 2, 3, 4 | 3eqtr3i 2640 | 1 ⊢ (𝐴 × ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∅c0 3874 × cxp 5036 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 |
This theorem is referenced by: xpnz 5472 xpdisj2 5475 difxp1 5478 dmxpss 5484 rnxpid 5486 xpcan 5489 unixp 5585 fconst5 6376 dfac5lem3 8831 xpcdaen 8888 fpwwe2lem13 9343 comfffval 16181 0ssc 16320 fuchom 16444 xpccofval 16645 frmdplusg 17214 mulgfval 17365 mulgfvi 17368 ga0 17554 symgplusg 17632 efgval 17953 psrplusg 19202 psrvscafval 19211 opsrle 19296 ply1plusgfvi 19433 txindislem 21246 txhaus 21260 0met 21981 aciunf1 28845 mbfmcst 29648 0rrv 29840 mexval 30653 mdvval 30655 mpstval 30686 dfpo2 30898 elima4 30924 finxp00 32415 isbnd3 32753 zrdivrng 32922 |
Copyright terms: Public domain | W3C validator |