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Mirrors > Home > ILE Home > Th. List > xpsnen2g | Unicode version |
Description: A set is equinumerous to its Cartesian product with a singleton on the left. (Contributed by Stefan O'Rear, 21-Nov-2014.) |
Ref | Expression |
---|---|
xpsnen2g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snexg 4145 | . . 3 | |
2 | xpcomeng 6773 | . . 3 | |
3 | 1, 2 | sylan 281 | . 2 |
4 | xpsneng 6767 | . . 3 | |
5 | 4 | ancoms 266 | . 2 |
6 | entr 6729 | . 2 | |
7 | 3, 5, 6 | syl2anc 409 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2128 cvv 2712 csn 3560 class class class wbr 3965 cxp 4584 cen 6683 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1st 6088 df-2nd 6089 df-er 6480 df-en 6686 |
This theorem is referenced by: djucomen 7151 djuassen 7152 xpdjuen 7153 |
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