| 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 5784 | . . 3 ⊢ (∅ × 𝐴) = ∅ | |
| 2 | 1 | cnveqi 5885 | . 2 ⊢ ◡(∅ × 𝐴) = ◡∅ |
| 3 | cnvxp 6177 | . 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅) | |
| 4 | cnv0 6160 | . 2 ⊢ ◡∅ = ∅ | |
| 5 | 2, 3, 4 | 3eqtr3i 2773 | 1 ⊢ (𝐴 × ∅) = ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∅c0 4333 × cxp 5683 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 |
| This theorem is referenced by: xpnz 6179 xpdisj2 6182 difxp1 6185 dmxpss 6191 rnxpid 6193 xpcan 6196 unixp 6302 dfpo2 6316 fconst5 7226 dfac5lem3 10165 djuassen 10219 xpdjuen 10220 alephadd 10617 fpwwe2lem12 10682 0ssc 17882 fuchom 18009 frmdplusg 18867 mulgfval 19087 mulgfvalALT 19088 mulgfvi 19091 ga0 19316 efgval 19735 psrplusg 21956 psrvscafval 21968 opsrle 22065 ply1plusgfvi 22243 txindislem 23641 txhaus 23655 0met 24376 2ndimaxp 32656 aciunf1 32673 hashxpe 32811 mbfmcst 34261 0rrv 34453 sate0 35420 mexval 35507 mdvval 35509 mpstval 35540 elima4 35776 finxp00 37403 isbnd3 37791 zrdivrng 37960 mofeu 48757 fucofvalne 49020 |
| Copyright terms: Public domain | W3C validator |