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Mirrors > Home > ILE Home > Th. List > fientri3 | Unicode version |
Description: Trichotomy of dominance for finite sets. (Contributed by Jim Kingdon, 15-Sep-2021.) |
Ref | Expression |
---|---|
fientri3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isfi 6623 | . . . 4 | |
2 | 1 | biimpi 119 | . . 3 |
3 | 2 | adantr 274 | . 2 |
4 | isfi 6623 | . . . . 5 | |
5 | 4 | biimpi 119 | . . . 4 |
6 | 5 | ad2antlr 480 | . . 3 |
7 | simplrr 510 | . . . . . . . 8 | |
8 | 7 | adantr 274 | . . . . . . 7 |
9 | simpr 109 | . . . . . . . 8 | |
10 | simplrl 509 | . . . . . . . . . 10 | |
11 | 10 | adantr 274 | . . . . . . . . 9 |
12 | simplrl 509 | . . . . . . . . 9 | |
13 | nndomo 6726 | . . . . . . . . 9 | |
14 | 11, 12, 13 | syl2anc 408 | . . . . . . . 8 |
15 | 9, 14 | mpbird 166 | . . . . . . 7 |
16 | endomtr 6652 | . . . . . . 7 | |
17 | 8, 15, 16 | syl2anc 408 | . . . . . 6 |
18 | simplrr 510 | . . . . . . 7 | |
19 | 18 | ensymd 6645 | . . . . . 6 |
20 | domentr 6653 | . . . . . 6 | |
21 | 17, 19, 20 | syl2anc 408 | . . . . 5 |
22 | 21 | orcd 707 | . . . 4 |
23 | simplrr 510 | . . . . . . 7 | |
24 | simpr 109 | . . . . . . . 8 | |
25 | simplrl 509 | . . . . . . . . 9 | |
26 | 10 | adantr 274 | . . . . . . . . 9 |
27 | nndomo 6726 | . . . . . . . . 9 | |
28 | 25, 26, 27 | syl2anc 408 | . . . . . . . 8 |
29 | 24, 28 | mpbird 166 | . . . . . . 7 |
30 | endomtr 6652 | . . . . . . 7 | |
31 | 23, 29, 30 | syl2anc 408 | . . . . . 6 |
32 | 7 | adantr 274 | . . . . . . 7 |
33 | 32 | ensymd 6645 | . . . . . 6 |
34 | domentr 6653 | . . . . . 6 | |
35 | 31, 33, 34 | syl2anc 408 | . . . . 5 |
36 | 35 | olcd 708 | . . . 4 |
37 | simprl 505 | . . . . 5 | |
38 | nntri2or2 6362 | . . . . 5 | |
39 | 10, 37, 38 | syl2anc 408 | . . . 4 |
40 | 22, 36, 39 | mpjaodan 772 | . . 3 |
41 | 6, 40 | rexlimddv 2531 | . 2 |
42 | 3, 41 | rexlimddv 2531 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 682 wcel 1465 wrex 2394 wss 3041 class class class wbr 3899 com 4474 cen 6600 cdom 6601 cfn 6602 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-br 3900 df-opab 3960 df-tr 3997 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-er 6397 df-en 6603 df-dom 6604 df-fin 6605 |
This theorem is referenced by: (None) |
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