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Mirrors > Home > MPE Home > Th. List > supex | Structured version Visualization version GIF version |
Description: A supremum is a set. (Contributed by NM, 22-May-1999.) |
Ref | Expression |
---|---|
supex.1 | ⊢ 𝑅 Or 𝐴 |
Ref | Expression |
---|---|
supex | ⊢ sup(𝐵, 𝐴, 𝑅) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supex.1 | . 2 ⊢ 𝑅 Or 𝐴 | |
2 | id 22 | . . 3 ⊢ (𝑅 Or 𝐴 → 𝑅 Or 𝐴) | |
3 | 2 | supexd 8905 | . 2 ⊢ (𝑅 Or 𝐴 → sup(𝐵, 𝐴, 𝑅) ∈ V) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ sup(𝐵, 𝐴, 𝑅) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Vcvv 3492 Or wor 5466 supcsup 8892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rmo 3143 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-po 5467 df-so 5468 df-sup 8894 |
This theorem is referenced by: limsupgval 14821 limsupgre 14826 gcdval 15833 pczpre 16172 prmreclem1 16240 prdsdsfn 16726 prdsdsval 16739 xrge0tsms2 23370 mbfsup 24192 mbfinf 24193 itg2val 24256 itg2monolem1 24278 itg2mono 24281 mdegval 24584 mdegxrf 24589 plyeq0lem 24727 dgrval 24745 nmooval 28467 nmopval 29560 nmfnval 29580 lmdvg 31095 esumval 31204 erdszelem3 32337 erdszelem6 32340 supcnvlimsup 41897 limsuplt2 41910 liminfval 41916 limsupge 41918 liminflelimsuplem 41932 fourierdlem79 42347 sge0val 42525 sge0tsms 42539 smflimsuplem1 42971 smflimsuplem2 42972 smflimsuplem4 42974 |
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