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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | endom 7614 |
. . 3
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2 | sdomnen 7616 |
. . . 4
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3 | 2 | con2i 124 |
. . 3
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4 | 1, 3 | jca 541 |
. 2
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5 | brdom2 7617 |
. . . 4
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6 | 5 | biimpi 199 |
. . 3
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7 | 6 | orcanai 927 |
. 2
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8 | 4, 7 | impbii 192 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-br 4396 df-opab 4455 df-xp 4845 df-rel 4846 df-f1o 5596 df-en 7588 df-dom 7589 df-sdom 7590 |
This theorem is referenced by: marypha1lem 7965 tskwe 8402 infxpenlem 8462 cdainflem 8639 axcclem 8905 alephsuc3 9023 gchen1 9068 gchen2 9069 inatsk 9221 ufilen 21023 dirith2 24445 f1ocnt 28451 mblfinlem1 32041 |
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