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Mirrors > Home > MPE Home > Th. List > eqsup | Structured version Visualization version Unicode version |
Description: Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.) |
Ref | Expression |
---|---|
supmo.1 |
Ref | Expression |
---|---|
eqsup |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supmo.1 | . . . . 5 | |
2 | 1 | adantr 481 | . . . 4 |
3 | 2 | supval2 8361 | . . 3 |
4 | 3simpc 1060 | . . . . 5 | |
5 | 4 | adantl 482 | . . . 4 |
6 | simpr1 1067 | . . . . 5 | |
7 | breq1 4656 | . . . . . . . . . . 11 | |
8 | 7 | notbid 308 | . . . . . . . . . 10 |
9 | 8 | ralbidv 2986 | . . . . . . . . 9 |
10 | breq2 4657 | . . . . . . . . . . 11 | |
11 | 10 | imbi1d 331 | . . . . . . . . . 10 |
12 | 11 | ralbidv 2986 | . . . . . . . . 9 |
13 | 9, 12 | anbi12d 747 | . . . . . . . 8 |
14 | 13 | rspcev 3309 | . . . . . . 7 |
15 | 6, 5, 14 | syl2anc 693 | . . . . . 6 |
16 | 2, 15 | supeu 8360 | . . . . 5 |
17 | 13 | riota2 6633 | . . . . 5 |
18 | 6, 16, 17 | syl2anc 693 | . . . 4 |
19 | 5, 18 | mpbid 222 | . . 3 |
20 | 3, 19 | eqtrd 2656 | . 2 |
21 | 20 | ex 450 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wrex 2913 wreu 2914 class class class wbr 4653 wor 5034 crio 6610 csup 8346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-po 5035 df-so 5036 df-iota 5851 df-riota 6611 df-sup 8348 |
This theorem is referenced by: eqsupd 8363 eqinf 8390 suprzcl2 11778 supxr 12143 |
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