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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 3612 | . . 3 | |
4 | 2, 3 | syl 14 | . 2 |
5 | eqid 2170 | . . . . . 6 | |
6 | 1 | ralrimivva 2552 | . . . . . . 7 |
7 | eqeq1 2177 | . . . . . . . . . 10 | |
8 | breq1 3992 | . . . . . . . . . . . 12 | |
9 | 8 | notbid 662 | . . . . . . . . . . 11 |
10 | breq2 3993 | . . . . . . . . . . . 12 | |
11 | 10 | notbid 662 | . . . . . . . . . . 11 |
12 | 9, 11 | anbi12d 470 | . . . . . . . . . 10 |
13 | 7, 12 | bibi12d 234 | . . . . . . . . 9 |
14 | eqeq2 2180 | . . . . . . . . . 10 | |
15 | breq2 3993 | . . . . . . . . . . . 12 | |
16 | 15 | notbid 662 | . . . . . . . . . . 11 |
17 | breq1 3992 | . . . . . . . . . . . 12 | |
18 | 17 | notbid 662 | . . . . . . . . . . 11 |
19 | 16, 18 | anbi12d 470 | . . . . . . . . . 10 |
20 | 14, 19 | bibi12d 234 | . . . . . . . . 9 |
21 | 13, 20 | rspc2v 2847 | . . . . . . . 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 3601 | . . . . . 6 | |
28 | 27 | breq2d 4001 | . . . . 5 |
29 | 28 | notbid 662 | . . . 4 |
30 | 29 | adantl 275 | . . 3 |
31 | 26, 30 | mpbird 166 | . 2 |
32 | 1, 2, 4, 31 | supmaxti 6981 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wral 2448 csn 3583 class class class wbr 3989 csup 6959 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-riota 5809 df-sup 6961 |
This theorem is referenced by: infsnti 7007 |
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