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Mirrors > Home > ILE Home > Th. List > fodjuomni | Unicode version |
Description: A condition which ensures is either inhabited or empty. Lemma 3.2 of [PradicBrown2022], p. 4. (Contributed by Jim Kingdon, 27-Jul-2022.) |
Ref | Expression |
---|---|
fodjuomni.o | Omni |
fodjuomni.fo | ⊔ |
Ref | Expression |
---|---|
fodjuomni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodjuomni.o | . 2 Omni | |
2 | fodjuomni.fo | . 2 ⊔ | |
3 | fveq2 5467 | . . . . . . 7 inl inl | |
4 | 3 | eqeq2d 2169 | . . . . . 6 inl inl |
5 | 4 | cbvrexv 2681 | . . . . 5 inl inl |
6 | ifbi 3525 | . . . . 5 inl inl inl inl | |
7 | 5, 6 | ax-mp 5 | . . . 4 inl inl |
8 | 7 | mpteq2i 4051 | . . 3 inl inl |
9 | fveq2 5467 | . . . . . . 7 | |
10 | 9 | eqeq1d 2166 | . . . . . 6 inl inl |
11 | 10 | rexbidv 2458 | . . . . 5 inl inl |
12 | 11 | ifbid 3526 | . . . 4 inl inl |
13 | 12 | cbvmptv 4060 | . . 3 inl inl |
14 | 8, 13 | eqtri 2178 | . 2 inl inl |
15 | 1, 2, 14 | fodjuomnilemres 7086 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wo 698 wceq 1335 wex 1472 wcel 2128 wrex 2436 c0 3394 cif 3505 cmpt 4025 wfo 5167 cfv 5169 c1o 6353 ⊔ cdju 6976 inlcinl 6984 Omnicomni 7072 |
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 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 |
This theorem depends on definitions: df-bi 116 df-dc 821 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-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-1o 6360 df-2o 6361 df-map 6592 df-dju 6977 df-inl 6986 df-inr 6987 df-omni 7073 |
This theorem is referenced by: ctssexmid 7088 exmidunben 12142 exmidsbthrlem 13580 sbthomlem 13583 |
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