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Mirrors > Home > ILE Home > Th. List > fodju0 | Unicode version |
Description: Lemma for fodjuomni 6989 and fodjumkv 7002. A condition which shows that is empty. (Contributed by Jim Kingdon, 27-Jul-2022.) (Revised by Jim Kingdon, 25-Mar-2023.) |
Ref | Expression |
---|---|
fodjuf.fo | ⊔ |
fodjuf.p | inl |
fodju0.1 |
Ref | Expression |
---|---|
fodju0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodjuf.fo | . . . . 5 ⊔ | |
2 | djulcl 6904 | . . . . 5 inl ⊔ | |
3 | foelrn 5622 | . . . . 5 ⊔ inl ⊔ inl | |
4 | 1, 2, 3 | syl2an 287 | . . . 4 inl |
5 | fodjuf.p | . . . . . 6 inl | |
6 | fveqeq2 5398 | . . . . . . . 8 inl inl | |
7 | 6 | rexbidv 2415 | . . . . . . 7 inl inl |
8 | 7 | ifbid 3463 | . . . . . 6 inl inl |
9 | simprl 505 | . . . . . 6 inl | |
10 | peano1 4478 | . . . . . . . 8 | |
11 | 10 | a1i 9 | . . . . . . 7 inl |
12 | 1onn 6384 | . . . . . . . 8 | |
13 | 12 | a1i 9 | . . . . . . 7 inl |
14 | 1 | fodjuomnilemdc 6984 | . . . . . . . 8 DECID inl |
15 | 14 | ad2ant2r 500 | . . . . . . 7 inl DECID inl |
16 | 11, 13, 15 | ifcldcd 3477 | . . . . . 6 inl inl |
17 | 5, 8, 9, 16 | fvmptd3 5482 | . . . . 5 inl inl |
18 | fveqeq2 5398 | . . . . . 6 | |
19 | fodju0.1 | . . . . . . 7 | |
20 | 19 | ad2antrr 479 | . . . . . 6 inl |
21 | 18, 20, 9 | rspcdva 2768 | . . . . 5 inl |
22 | simplr 504 | . . . . . . 7 inl | |
23 | simprr 506 | . . . . . . . 8 inl inl | |
24 | 23 | eqcomd 2123 | . . . . . . 7 inl inl |
25 | fveq2 5389 | . . . . . . . 8 inl inl | |
26 | 25 | rspceeqv 2781 | . . . . . . 7 inl inl |
27 | 22, 24, 26 | syl2anc 408 | . . . . . 6 inl inl |
28 | 27 | iftrued 3451 | . . . . 5 inl inl |
29 | 17, 21, 28 | 3eqtr3rd 2159 | . . . 4 inl |
30 | 4, 29 | rexlimddv 2531 | . . 3 |
31 | 1n0 6297 | . . . . 5 | |
32 | 31 | nesymi 2331 | . . . 4 |
33 | 32 | a1i 9 | . . 3 |
34 | 30, 33 | pm2.65da 635 | . 2 |
35 | 34 | eq0rdv 3377 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 DECID wdc 804 wceq 1316 wcel 1465 wral 2393 wrex 2394 c0 3333 cif 3444 cmpt 3959 com 4474 wfo 5091 cfv 5093 c1o 6274 ⊔ cdju 6890 inlcinl 6898 |
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 |
This theorem depends on definitions: df-bi 116 df-dc 805 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-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 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-mpt 3961 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-1st 6006 df-2nd 6007 df-1o 6281 df-dju 6891 df-inl 6900 df-inr 6901 |
This theorem is referenced by: fodjuomnilemres 6988 fodjumkvlemres 7001 |
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