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Mirrors > Home > MPE Home > Th. List > fidomtri | Structured version Visualization version Unicode version |
Description: Trichotomy of dominance without AC when one set is finite. (Contributed by Stefan O'Rear, 30-Oct-2014.) (Revised by Mario Carneiro, 27-Apr-2015.) |
Ref | Expression |
---|---|
fidomtri |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnsym 8086 | . 2 | |
2 | finnum 8774 | . . . . . 6 | |
3 | 2 | adantr 481 | . . . . 5 |
4 | finnum 8774 | . . . . 5 | |
5 | domtri2 8815 | . . . . 5 | |
6 | 3, 4, 5 | syl2an 494 | . . . 4 |
7 | 6 | biimprd 238 | . . 3 |
8 | isinffi 8818 | . . . . . . 7 | |
9 | 8 | ancoms 469 | . . . . . 6 |
10 | 9 | adantlr 751 | . . . . 5 |
11 | brdomg 7965 | . . . . . 6 | |
12 | 11 | ad2antlr 763 | . . . . 5 |
13 | 10, 12 | mpbird 247 | . . . 4 |
14 | 13 | a1d 25 | . . 3 |
15 | 7, 14 | pm2.61dan 832 | . 2 |
16 | 1, 15 | impbid2 216 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wex 1704 wcel 1990 class class class wbr 4653 cdm 5114 wf1 5885 cdom 7953 csdm 7954 cfn 7955 ccrd 8761 |
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-8 1992 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-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 |
This theorem is referenced by: fidomtri2 8820 fin56 9215 hauspwdom 21304 harinf 37601 |
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