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Mirrors > Home > ILE Home > Th. List > ssfilem | Unicode version |
Description: Lemma for ssfiexmid 6763. (Contributed by Jim Kingdon, 3-Feb-2022.) |
Ref | Expression |
---|---|
ssfilem.1 |
Ref | Expression |
---|---|
ssfilem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssfilem.1 | . . 3 | |
2 | isfi 6648 | . . 3 | |
3 | 1, 2 | mpbi 144 | . 2 |
4 | 0elnn 4527 | . . . . 5 | |
5 | breq2 3928 | . . . . . . . . . 10 | |
6 | en0 6682 | . . . . . . . . . 10 | |
7 | 5, 6 | syl6bb 195 | . . . . . . . . 9 |
8 | 7 | biimpac 296 | . . . . . . . 8 |
9 | rabeq0 3387 | . . . . . . . . 9 | |
10 | 0ex 4050 | . . . . . . . . . . 11 | |
11 | 10 | snm 3638 | . . . . . . . . . 10 |
12 | r19.3rmv 3448 | . . . . . . . . . 10 | |
13 | 11, 12 | ax-mp 5 | . . . . . . . . 9 |
14 | 9, 13 | bitr4i 186 | . . . . . . . 8 |
15 | 8, 14 | sylib 121 | . . . . . . 7 |
16 | 15 | olcd 723 | . . . . . 6 |
17 | ensym 6668 | . . . . . . . 8 | |
18 | elex2 2697 | . . . . . . . 8 | |
19 | enm 6707 | . . . . . . . 8 | |
20 | 17, 18, 19 | syl2an 287 | . . . . . . 7 |
21 | biidd 171 | . . . . . . . . . . 11 | |
22 | 21 | elrab 2835 | . . . . . . . . . 10 |
23 | 22 | simprbi 273 | . . . . . . . . 9 |
24 | 23 | orcd 722 | . . . . . . . 8 |
25 | 24 | exlimiv 1577 | . . . . . . 7 |
26 | 20, 25 | syl 14 | . . . . . 6 |
27 | 16, 26 | jaodan 786 | . . . . 5 |
28 | 4, 27 | sylan2 284 | . . . 4 |
29 | 28 | ancoms 266 | . . 3 |
30 | 29 | rexlimiva 2542 | . 2 |
31 | 3, 30 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wa 103 wb 104 wo 697 wceq 1331 wex 1468 wcel 1480 wral 2414 wrex 2415 crab 2418 c0 3358 csn 3522 class class class wbr 3924 com 4499 cen 6625 cfn 6627 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-er 6422 df-en 6628 df-fin 6630 |
This theorem is referenced by: ssfiexmid 6763 domfiexmid 6765 |
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