Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > idssen | Unicode version |
Description: Equality implies equinumerosity. (Contributed by NM, 30-Apr-1998.) (Revised by Mario Carneiro, 15-Nov-2014.) |
Ref | Expression |
---|---|
idssen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reli 5135 | . 2 | |
2 | vex 3112 | . . . . 5 | |
3 | 2 | ideq 5160 | . . . 4 |
4 | vex 3112 | . . . . 5 | |
5 | eqeng 7569 | . . . . 5 | |
6 | 4, 5 | ax-mp 5 | . . . 4 |
7 | 3, 6 | sylbi 195 | . . 3 |
8 | df-br 4453 | . . 3 | |
9 | df-br 4453 | . . 3 | |
10 | 7, 8, 9 | 3imtr3i 265 | . 2 |
11 | 1, 10 | relssi 5099 | 1 |
Colors of variables: wff setvar class |
Syntax hints: -> wi 4 e. wcel 1818
cvv 3109
C_ wss 3475 <. cop 4035 class class class wbr 4452
cid 4795
cen 7533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-8 1820 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pow 4630 ax-pr 4691 ax-un 6592 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-id 4800 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 df-ima 5017 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600 df-en 7537 |
Copyright terms: Public domain | W3C validator |