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Mirrors > Home > ILE Home > Th. List > ssfilem | Unicode version |
Description: Lemma for ssfiexmid 6823. (Contributed by Jim Kingdon, 3-Feb-2022.) |
Ref | Expression |
---|---|
ssfilem.1 |
Ref | Expression |
---|---|
ssfilem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssfilem.1 | . . 3 | |
2 | isfi 6708 | . . 3 | |
3 | 1, 2 | mpbi 144 | . 2 |
4 | 0elnn 4580 | . . . . 5 | |
5 | breq2 3971 | . . . . . . . . . 10 | |
6 | en0 6742 | . . . . . . . . . 10 | |
7 | 5, 6 | bitrdi 195 | . . . . . . . . 9 |
8 | 7 | biimpac 296 | . . . . . . . 8 |
9 | rabeq0 3424 | . . . . . . . . 9 | |
10 | 0ex 4093 | . . . . . . . . . . 11 | |
11 | 10 | snm 3681 | . . . . . . . . . 10 |
12 | r19.3rmv 3485 | . . . . . . . . . 10 | |
13 | 11, 12 | ax-mp 5 | . . . . . . . . 9 |
14 | 9, 13 | bitr4i 186 | . . . . . . . 8 |
15 | 8, 14 | sylib 121 | . . . . . . 7 |
16 | 15 | olcd 724 | . . . . . 6 |
17 | ensym 6728 | . . . . . . . 8 | |
18 | elex2 2728 | . . . . . . . 8 | |
19 | enm 6767 | . . . . . . . 8 | |
20 | 17, 18, 19 | syl2an 287 | . . . . . . 7 |
21 | biidd 171 | . . . . . . . . . . 11 | |
22 | 21 | elrab 2868 | . . . . . . . . . 10 |
23 | 22 | simprbi 273 | . . . . . . . . 9 |
24 | 23 | orcd 723 | . . . . . . . 8 |
25 | 24 | exlimiv 1578 | . . . . . . 7 |
26 | 20, 25 | syl 14 | . . . . . 6 |
27 | 16, 26 | jaodan 787 | . . . . 5 |
28 | 4, 27 | sylan2 284 | . . . 4 |
29 | 28 | ancoms 266 | . . 3 |
30 | 29 | rexlimiva 2569 | . 2 |
31 | 3, 30 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 698 wceq 1335 wex 1472 wcel 2128 wral 2435 wrex 2436 crab 2439 c0 3395 csn 3561 class class class wbr 3967 com 4551 cen 6685 cfn 6687 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4084 ax-nul 4092 ax-pow 4137 ax-pr 4171 ax-un 4395 ax-iinf 4549 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4028 df-id 4255 df-suc 4333 df-iom 4552 df-xp 4594 df-rel 4595 df-cnv 4596 df-co 4597 df-dm 4598 df-rn 4599 df-res 4600 df-ima 4601 df-iota 5137 df-fun 5174 df-fn 5175 df-f 5176 df-f1 5177 df-fo 5178 df-f1o 5179 df-fv 5180 df-er 6482 df-en 6688 df-fin 6690 |
This theorem is referenced by: ssfiexmid 6823 domfiexmid 6825 |
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