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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 8359 | . 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 1990 Vcvv 3200 Or wor 5034 supcsup 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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
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-ral 2917 df-rex 2918 df-rmo 2920 df-rab 2921 df-v 3202 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-sup 8348 |
This theorem is referenced by: limsupgval 14207 limsupgre 14212 gcdval 15218 pczpre 15552 prmreclem1 15620 prdsdsfn 16125 prdsdsval 16138 xrge0tsms2 22638 mbfsup 23431 mbfinf 23432 itg2val 23495 itg2monolem1 23517 itg2mono 23520 mdegval 23823 mdegxrf 23828 plyeq0lem 23966 dgrval 23984 nmooval 27618 nmopval 28715 nmfnval 28735 lmdvg 29999 esumval 30108 erdszelem3 31175 erdszelem6 31178 gtinfOLD 32314 supcnvlimsup 39972 limsuplt2 39985 liminfval 39991 limsupge 39993 liminflelimsuplem 40007 fourierdlem79 40402 sge0val 40583 sge0tsms 40597 smflimsuplem1 41026 smflimsuplem2 41027 smflimsuplem4 41029 |
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