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Mirrors > Home > MPE Home > Th. List > bren2 | Structured version Visualization version Unicode version |
Description: Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.) |
Ref | Expression |
---|---|
bren2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endom 7982 | . . 3 | |
2 | sdomnen 7984 | . . . 4 | |
3 | 2 | con2i 134 | . . 3 |
4 | 1, 3 | jca 554 | . 2 |
5 | brdom2 7985 | . . . 4 | |
6 | 5 | biimpi 206 | . . 3 |
7 | 6 | orcanai 952 | . 2 |
8 | 4, 7 | impbii 199 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 196 wo 383 wa 384 class class class wbr 4653 cen 7952 cdom 7953 csdm 7954 |
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-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 |
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-ral 2917 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-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-f1o 5895 df-en 7956 df-dom 7957 df-sdom 7958 |
This theorem is referenced by: marypha1lem 8339 tskwe 8776 infxpenlem 8836 cdainflem 9013 axcclem 9279 alephsuc3 9402 gchen1 9447 gchen2 9448 inatsk 9600 ufilen 21734 dirith2 25217 f1ocnt 29559 lindsenlbs 33404 mblfinlem1 33446 axccdom 39416 axccd2 39430 |
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