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Mirrors > Home > MPE Home > Th. List > infex | Structured version Visualization version GIF version |
Description: An infimum is a set. (Contributed by AV, 3-Sep-2020.) |
Ref | Expression |
---|---|
infex.1 | ⊢ 𝑅 Or 𝐴 |
Ref | Expression |
---|---|
infex | ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infex.1 | . 2 ⊢ 𝑅 Or 𝐴 | |
2 | id 22 | . . 3 ⊢ (𝑅 Or 𝐴 → 𝑅 Or 𝐴) | |
3 | 2 | infexd 9474 | . 2 ⊢ (𝑅 Or 𝐴 → inf(𝐵, 𝐴, 𝑅) ∈ V) |
4 | 1, 3 | ax-mp 5 | 1 ⊢ inf(𝐵, 𝐴, 𝑅) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 Vcvv 3475 Or wor 5586 infcinf 9432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-po 5587 df-so 5588 df-cnv 5683 df-sup 9433 df-inf 9434 |
This theorem is referenced by: limsupval 15414 lcmval 16525 odzval 16720 ramval 16937 imasdsfn 17456 imasdsval 17457 odval 19395 odf 19398 gexval 19439 nmoval 24214 metdsval 24345 ovolval 24972 ovolf 24981 elqaalem1 25814 elqaalem3 25816 ballotlemi 33437 pellfundval 41551 dgraaval 41819 dgraaf 41822 liminfgval 44413 liminfval2 44419 ovnval2 45196 finfdm2 45498 |
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