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Mirrors > Home > ILE Home > Th. List > 2dom | Unicode version |
Description: A set that dominates ordinal 2 has at least 2 different members. (Contributed by NM, 25-Jul-2004.) |
Ref | Expression |
---|---|
2dom |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df2o2 6015 | . . . 4 | |
2 | 1 | breq1i 3771 | . . 3 |
3 | brdomi 6230 | . . 3 | |
4 | 2, 3 | sylbi 114 | . 2 |
5 | f1f 5092 | . . . . 5 | |
6 | 0ex 3884 | . . . . . 6 | |
7 | 6 | prid1 3476 | . . . . 5 |
8 | ffvelrn 5300 | . . . . 5 | |
9 | 5, 7, 8 | sylancl 392 | . . . 4 |
10 | p0ex 3939 | . . . . . 6 | |
11 | 10 | prid2 3477 | . . . . 5 |
12 | ffvelrn 5300 | . . . . 5 | |
13 | 5, 11, 12 | sylancl 392 | . . . 4 |
14 | 0nep0 3918 | . . . . . 6 | |
15 | 14 | neii 2208 | . . . . 5 |
16 | f1fveq 5411 | . . . . . 6 | |
17 | 7, 11, 16 | mpanr12 415 | . . . . 5 |
18 | 15, 17 | mtbiri 600 | . . . 4 |
19 | eqeq1 2046 | . . . . . 6 | |
20 | 19 | notbid 592 | . . . . 5 |
21 | eqeq2 2049 | . . . . . 6 | |
22 | 21 | notbid 592 | . . . . 5 |
23 | 20, 22 | rspc2ev 2664 | . . . 4 |
24 | 9, 13, 18, 23 | syl3anc 1135 | . . 3 |
25 | 24 | exlimiv 1489 | . 2 |
26 | 4, 25 | syl 14 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wb 98 wceq 1243 wex 1381 wcel 1393 wrex 2307 c0 3224 csn 3375 cpr 3376 class class class wbr 3764 wf 4898 wf1 4899 cfv 4902 c2o 5995 cdom 6220 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-suc 4108 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fv 4910 df-1o 6001 df-2o 6002 df-dom 6223 |
This theorem is referenced by: (None) |
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