Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > supsnti | Unicode version |
Description: The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.) |
Ref | Expression |
---|---|
supsnti.ti | |
supsnti.b |
Ref | Expression |
---|---|
supsnti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supsnti.ti | . 2 | |
2 | supsnti.b | . 2 | |
3 | snidg 3599 | . . 3 | |
4 | 2, 3 | syl 14 | . 2 |
5 | eqid 2164 | . . . . . 6 | |
6 | 1 | ralrimivva 2546 | . . . . . . 7 |
7 | eqeq1 2171 | . . . . . . . . . 10 | |
8 | breq1 3979 | . . . . . . . . . . . 12 | |
9 | 8 | notbid 657 | . . . . . . . . . . 11 |
10 | breq2 3980 | . . . . . . . . . . . 12 | |
11 | 10 | notbid 657 | . . . . . . . . . . 11 |
12 | 9, 11 | anbi12d 465 | . . . . . . . . . 10 |
13 | 7, 12 | bibi12d 234 | . . . . . . . . 9 |
14 | eqeq2 2174 | . . . . . . . . . 10 | |
15 | breq2 3980 | . . . . . . . . . . . 12 | |
16 | 15 | notbid 657 | . . . . . . . . . . 11 |
17 | breq1 3979 | . . . . . . . . . . . 12 | |
18 | 17 | notbid 657 | . . . . . . . . . . 11 |
19 | 16, 18 | anbi12d 465 | . . . . . . . . . 10 |
20 | 14, 19 | bibi12d 234 | . . . . . . . . 9 |
21 | 13, 20 | rspc2v 2838 | . . . . . . . 8 |
22 | 2, 2, 21 | syl2anc 409 | . . . . . . 7 |
23 | 6, 22 | mpd 13 | . . . . . 6 |
24 | 5, 23 | mpbii 147 | . . . . 5 |
25 | 24 | simpld 111 | . . . 4 |
26 | 25 | adantr 274 | . . 3 |
27 | elsni 3588 | . . . . . 6 | |
28 | 27 | breq2d 3988 | . . . . 5 |
29 | 28 | notbid 657 | . . . 4 |
30 | 29 | adantl 275 | . . 3 |
31 | 26, 30 | mpbird 166 | . 2 |
32 | 1, 2, 4, 31 | supmaxti 6960 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1342 wcel 2135 wral 2442 csn 3570 class class class wbr 3976 csup 6938 |
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-ext 2146 |
This theorem depends on definitions: df-bi 116 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-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-iota 5147 df-riota 5792 df-sup 6940 |
This theorem is referenced by: infsnti 6986 |
Copyright terms: Public domain | W3C validator |