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Mirrors > Home > ILE Home > Th. List > fodjuomnilemdc | Unicode version |
Description: Lemma for fodjuomni 7021. Decidability of a condition we use in various lemmas. (Contributed by Jim Kingdon, 27-Jul-2022.) |
Ref | Expression |
---|---|
fodjuomnilemdc.fo | ⊔ |
Ref | Expression |
---|---|
fodjuomnilemdc | DECID inl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodjuomnilemdc.fo | . . . . . 6 ⊔ | |
2 | fof 5345 | . . . . . 6 ⊔ ⊔ | |
3 | 1, 2 | syl 14 | . . . . 5 ⊔ |
4 | 3 | ffvelrnda 5555 | . . . 4 ⊔ |
5 | djur 6954 | . . . 4 ⊔ inl inr | |
6 | 4, 5 | sylib 121 | . . 3 inl inr |
7 | nfv 1508 | . . . . . . . 8 | |
8 | nfre1 2476 | . . . . . . . 8 inr | |
9 | 7, 8 | nfan 1544 | . . . . . . 7 inr |
10 | simpr 109 | . . . . . . . . . 10 inr inr | |
11 | fveq2 5421 | . . . . . . . . . . . 12 inr inr | |
12 | 11 | eqeq2d 2151 | . . . . . . . . . . 11 inr inr |
13 | 12 | cbvrexv 2655 | . . . . . . . . . 10 inr inr |
14 | 10, 13 | sylib 121 | . . . . . . . . 9 inr inr |
15 | vex 2689 | . . . . . . . . . . . . . . 15 | |
16 | vex 2689 | . . . . . . . . . . . . . . 15 | |
17 | djune 6963 | . . . . . . . . . . . . . . 15 inl inr | |
18 | 15, 16, 17 | mp2an 422 | . . . . . . . . . . . . . 14 inl inr |
19 | neeq2 2322 | . . . . . . . . . . . . . 14 inr inl inl inr | |
20 | 18, 19 | mpbiri 167 | . . . . . . . . . . . . 13 inr inl |
21 | 20 | necomd 2394 | . . . . . . . . . . . 12 inr inl |
22 | 21 | neneqd 2329 | . . . . . . . . . . 11 inr inl |
23 | 22 | a1i 9 | . . . . . . . . . 10 inr inr inl |
24 | 23 | rexlimdvw 2553 | . . . . . . . . 9 inr inr inl |
25 | 14, 24 | mpd 13 | . . . . . . . 8 inr inl |
26 | 25 | a1d 22 | . . . . . . 7 inr inl |
27 | 9, 26 | ralrimi 2503 | . . . . . 6 inr inl |
28 | ralnex 2426 | . . . . . 6 inl inl | |
29 | 27, 28 | sylib 121 | . . . . 5 inr inl |
30 | 29 | ex 114 | . . . 4 inr inl |
31 | 30 | orim2d 777 | . . 3 inl inr inl inl |
32 | 6, 31 | mpd 13 | . 2 inl inl |
33 | df-dc 820 | . 2 DECID inl inl inl | |
34 | 32, 33 | sylibr 133 | 1 DECID inl |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 697 DECID wdc 819 wceq 1331 wcel 1480 wne 2308 wral 2416 wrex 2417 cvv 2686 wf 5119 wfo 5121 cfv 5123 ⊔ cdju 6922 inlcinl 6930 inrcinr 6931 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1st 6038 df-2nd 6039 df-1o 6313 df-dju 6923 df-inl 6932 df-inr 6933 |
This theorem is referenced by: fodjuf 7017 fodjum 7018 fodju0 7019 |
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