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Mirrors > Home > ILE Home > Th. List > fodjumkv | Unicode version |
Description: A condition which ensures that a nonempty set is inhabited. (Contributed by Jim Kingdon, 25-Mar-2023.) |
Ref | Expression |
---|---|
fodjumkv.o | Markov |
fodjumkv.fo | ⊔ |
Ref | Expression |
---|---|
fodjumkv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fodjumkv.o | . 2 Markov | |
2 | fodjumkv.fo | . 2 ⊔ | |
3 | fveq2 5480 | . . . . . . 7 inl inl | |
4 | 3 | eqeq2d 2176 | . . . . . 6 inl inl |
5 | 4 | cbvrexv 2690 | . . . . 5 inl inl |
6 | ifbi 3535 | . . . . 5 inl inl inl inl | |
7 | 5, 6 | ax-mp 5 | . . . 4 inl inl |
8 | 7 | mpteq2i 4063 | . . 3 inl inl |
9 | fveqeq2 5489 | . . . . . 6 inl inl | |
10 | 9 | rexbidv 2465 | . . . . 5 inl inl |
11 | 10 | ifbid 3536 | . . . 4 inl inl |
12 | 11 | cbvmptv 4072 | . . 3 inl inl |
13 | 8, 12 | eqtri 2185 | . 2 inl inl |
14 | 1, 2, 13 | fodjumkvlemres 7114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1342 wex 1479 wcel 2135 wne 2334 wrex 2443 c0 3404 cif 3515 cmpt 4037 wfo 5180 cfv 5182 c1o 6368 ⊔ cdju 6993 inlcinl 7001 Markovcmarkov 7106 |
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 ax-setind 4508 |
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-rab 2451 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-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 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-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 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-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-1o 6375 df-2o 6376 df-map 6607 df-dju 6994 df-inl 7003 df-inr 7004 df-markov 7107 |
This theorem is referenced by: (None) |
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