Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > recexprlemell | Unicode version |
Description: Membership in the lower cut of . Lemma for recexpr 7558. (Contributed by Jim Kingdon, 27-Dec-2019.) |
Ref | Expression |
---|---|
recexpr.1 |
Ref | Expression |
---|---|
recexprlemell |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2723 | . 2 | |
2 | ltrelnq 7285 | . . . . . . 7 | |
3 | 2 | brel 4638 | . . . . . 6 |
4 | 3 | simpld 111 | . . . . 5 |
5 | elex 2723 | . . . . 5 | |
6 | 4, 5 | syl 14 | . . . 4 |
7 | 6 | adantr 274 | . . 3 |
8 | 7 | exlimiv 1578 | . 2 |
9 | breq1 3968 | . . . . 5 | |
10 | 9 | anbi1d 461 | . . . 4 |
11 | 10 | exbidv 1805 | . . 3 |
12 | recexpr.1 | . . . . 5 | |
13 | 12 | fveq2i 5471 | . . . 4 |
14 | nqex 7283 | . . . . . 6 | |
15 | 2 | brel 4638 | . . . . . . . . . 10 |
16 | 15 | simpld 111 | . . . . . . . . 9 |
17 | 16 | adantr 274 | . . . . . . . 8 |
18 | 17 | exlimiv 1578 | . . . . . . 7 |
19 | 18 | abssi 3203 | . . . . . 6 |
20 | 14, 19 | ssexi 4102 | . . . . 5 |
21 | 2 | brel 4638 | . . . . . . . . . 10 |
22 | 21 | simprd 113 | . . . . . . . . 9 |
23 | 22 | adantr 274 | . . . . . . . 8 |
24 | 23 | exlimiv 1578 | . . . . . . 7 |
25 | 24 | abssi 3203 | . . . . . 6 |
26 | 14, 25 | ssexi 4102 | . . . . 5 |
27 | 20, 26 | op1st 6094 | . . . 4 |
28 | 13, 27 | eqtri 2178 | . . 3 |
29 | 11, 28 | elab2g 2859 | . 2 |
30 | 1, 8, 29 | pm5.21nii 694 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wb 104 wceq 1335 wex 1472 wcel 2128 cab 2143 cvv 2712 cop 3563 class class class wbr 3965 cfv 5170 c1st 6086 c2nd 6087 cnq 7200 crq 7204 cltq 7205 |
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-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 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-ral 2440 df-rex 2441 df-reu 2442 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-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1st 6088 df-qs 6486 df-ni 7224 df-nqqs 7268 df-ltnqqs 7273 |
This theorem is referenced by: recexprlemm 7544 recexprlemopl 7545 recexprlemlol 7546 recexprlemdisj 7550 recexprlemloc 7551 recexprlem1ssl 7553 recexprlemss1l 7555 |
Copyright terms: Public domain | W3C validator |