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Mirrors > Home > ILE Home > Th. List > fodjuomnilemdc | Unicode version |
Description: Lemma for fodjuomni 7104. 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 5404 | . . . . . 6 ⊔ ⊔ | |
3 | 1, 2 | syl 14 | . . . . 5 ⊔ |
4 | 3 | ffvelrnda 5614 | . . . 4 ⊔ |
5 | djur 7025 | . . . 4 ⊔ inl inr | |
6 | 4, 5 | sylib 121 | . . 3 inl inr |
7 | nfv 1515 | . . . . . . . 8 | |
8 | nfre1 2507 | . . . . . . . 8 inr | |
9 | 7, 8 | nfan 1552 | . . . . . . 7 inr |
10 | simpr 109 | . . . . . . . . . 10 inr inr | |
11 | fveq2 5480 | . . . . . . . . . . . 12 inr inr | |
12 | 11 | eqeq2d 2176 | . . . . . . . . . . 11 inr inr |
13 | 12 | cbvrexv 2690 | . . . . . . . . . 10 inr inr |
14 | 10, 13 | sylib 121 | . . . . . . . . 9 inr inr |
15 | vex 2724 | . . . . . . . . . . . . . . 15 | |
16 | vex 2724 | . . . . . . . . . . . . . . 15 | |
17 | djune 7034 | . . . . . . . . . . . . . . 15 inl inr | |
18 | 15, 16, 17 | mp2an 423 | . . . . . . . . . . . . . 14 inl inr |
19 | neeq2 2348 | . . . . . . . . . . . . . 14 inr inl inl inr | |
20 | 18, 19 | mpbiri 167 | . . . . . . . . . . . . 13 inr inl |
21 | 20 | necomd 2420 | . . . . . . . . . . . 12 inr inl |
22 | 21 | neneqd 2355 | . . . . . . . . . . 11 inr inl |
23 | 22 | a1i 9 | . . . . . . . . . 10 inr inr inl |
24 | 23 | rexlimdvw 2585 | . . . . . . . . 9 inr inr inl |
25 | 14, 24 | mpd 13 | . . . . . . . 8 inr inl |
26 | 25 | a1d 22 | . . . . . . 7 inr inl |
27 | 9, 26 | ralrimi 2535 | . . . . . 6 inr inl |
28 | ralnex 2452 | . . . . . 6 inl inl | |
29 | 27, 28 | sylib 121 | . . . . 5 inr inl |
30 | 29 | ex 114 | . . . 4 inr inl |
31 | 30 | orim2d 778 | . . 3 inl inr inl inl |
32 | 6, 31 | mpd 13 | . 2 inl inl |
33 | df-dc 825 | . 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 698 DECID wdc 824 wceq 1342 wcel 2135 wne 2334 wral 2442 wrex 2443 cvv 2721 wf 5178 wfo 5180 cfv 5182 ⊔ cdju 6993 inlcinl 7001 inrcinr 7002 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-1st 6100 df-2nd 6101 df-1o 6375 df-dju 6994 df-inl 7003 df-inr 7004 |
This theorem is referenced by: fodjuf 7100 fodjum 7101 fodju0 7102 |
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