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Mirrors > Home > ILE Home > Th. List > recexprlemelu | Unicode version |
Description: Membership in the upper cut of . Lemma for recexpr 6736. (Contributed by Jim Kingdon, 27-Dec-2019.) |
Ref | Expression |
---|---|
recexpr.1 |
Ref | Expression |
---|---|
recexprlemelu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2566 | . 2 | |
2 | ltrelnq 6463 | . . . . . . 7 | |
3 | 2 | brel 4392 | . . . . . 6 |
4 | 3 | simprd 107 | . . . . 5 |
5 | elex 2566 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 6 | adantr 261 | . . 3 |
8 | 7 | exlimiv 1489 | . 2 |
9 | breq2 3768 | . . . . 5 | |
10 | 9 | anbi1d 438 | . . . 4 |
11 | 10 | exbidv 1706 | . . 3 |
12 | recexpr.1 | . . . . 5 | |
13 | 12 | fveq2i 5181 | . . . 4 |
14 | nqex 6461 | . . . . . 6 | |
15 | 2 | brel 4392 | . . . . . . . . . 10 |
16 | 15 | simpld 105 | . . . . . . . . 9 |
17 | 16 | adantr 261 | . . . . . . . 8 |
18 | 17 | exlimiv 1489 | . . . . . . 7 |
19 | 18 | abssi 3015 | . . . . . 6 |
20 | 14, 19 | ssexi 3895 | . . . . 5 |
21 | 2 | brel 4392 | . . . . . . . . . 10 |
22 | 21 | simprd 107 | . . . . . . . . 9 |
23 | 22 | adantr 261 | . . . . . . . 8 |
24 | 23 | exlimiv 1489 | . . . . . . 7 |
25 | 24 | abssi 3015 | . . . . . 6 |
26 | 14, 25 | ssexi 3895 | . . . . 5 |
27 | 20, 26 | op2nd 5774 | . . . 4 |
28 | 13, 27 | eqtri 2060 | . . 3 |
29 | 11, 28 | elab2g 2689 | . 2 |
30 | 1, 8, 29 | pm5.21nii 620 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cab 2026 cvv 2557 cop 3378 class class class wbr 3764 cfv 4902 c1st 5765 c2nd 5766 cnq 6378 crq 6382 cltq 6383 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-2nd 5768 df-qs 6112 df-ni 6402 df-nqqs 6446 df-ltnqqs 6451 |
This theorem is referenced by: recexprlemm 6722 recexprlemopu 6725 recexprlemupu 6726 recexprlemdisj 6728 recexprlemloc 6729 recexprlem1ssu 6732 recexprlemss1u 6734 |
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