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Mirrors > Home > MPE Home > Th. List > supsn | Structured version Visualization version Unicode version |
Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007.) |
Ref | Expression |
---|---|
supsn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4190 | . . . 4 | |
2 | 1 | supeq1i 8353 | . . 3 |
3 | suppr 8377 | . . . 4 | |
4 | 3 | 3anidm23 1385 | . . 3 |
5 | 2, 4 | syl5eq 2668 | . 2 |
6 | ifid 4125 | . 2 | |
7 | 5, 6 | syl6eq 2672 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cif 4086 csn 4177 cpr 4179 class class class wbr 4653 wor 5034 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: supxrmnf 12147 ramz 15729 xpsdsval 22186 ovolctb 23258 nmoo0 27646 nmop0 28845 nmfn0 28846 esumnul 30110 esum0 30111 ovoliunnfl 33451 voliunnfl 33453 volsupnfl 33454 liminf10ex 40006 fourierdlem79 40402 sge0z 40592 sge00 40593 |
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