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Mirrors > Home > ILE Home > Th. List > exmidfodomrlemreseldju | Unicode version |
Description: Lemma for exmidfodomrlemrALT 7059. A variant of eldju 6953. (Contributed by Jim Kingdon, 9-Jul-2022.) |
Ref | Expression |
---|---|
exmidfodomrlemreseldju.a | |
exmidfodomrlemreseldju.el | ⊔ |
Ref | Expression |
---|---|
exmidfodomrlemreseldju | inl inr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exmidfodomrlemreseldju.a | . . . . . . . . . . 11 | |
2 | 1 | sselda 3097 | . . . . . . . . . 10 |
3 | el1o 6334 | . . . . . . . . . 10 | |
4 | 2, 3 | sylib 121 | . . . . . . . . 9 |
5 | 4 | fveq2d 5425 | . . . . . . . 8 inl inl |
6 | 5 | eqeq2d 2151 | . . . . . . 7 inl inl |
7 | simpr 109 | . . . . . . . . 9 | |
8 | 4, 7 | eqeltrrd 2217 | . . . . . . . 8 |
9 | 8 | biantrurd 303 | . . . . . . 7 inl inl |
10 | 6, 9 | bitrd 187 | . . . . . 6 inl inl |
11 | 10 | biimpd 143 | . . . . 5 inl inl |
12 | 11 | rexlimdva 2549 | . . . 4 inl inl |
13 | 12 | imp 123 | . . 3 inl inl |
14 | 13 | orcd 722 | . 2 inl inl inr |
15 | simpr 109 | . . . . . . . . 9 | |
16 | 15, 3 | sylib 121 | . . . . . . . 8 |
17 | 16 | fveq2d 5425 | . . . . . . 7 inr inr |
18 | 17 | eqeq2d 2151 | . . . . . 6 inr inr |
19 | 18 | biimpd 143 | . . . . 5 inr inr |
20 | 19 | rexlimdva 2549 | . . . 4 inr inr |
21 | 20 | imp 123 | . . 3 inr inr |
22 | 21 | olcd 723 | . 2 inr inl inr |
23 | exmidfodomrlemreseldju.el | . . 3 ⊔ | |
24 | eldju 6953 | . . 3 ⊔ inl inr | |
25 | 23, 24 | sylib 121 | . 2 inl inr |
26 | 14, 22, 25 | mpjaodan 787 | 1 inl inr |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 697 wceq 1331 wcel 1480 wrex 2417 wss 3071 c0 3363 cres 4541 cfv 5123 c1o 6306 ⊔ 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-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 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: exmidfodomrlemrALT 7059 |
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