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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 9077 | . 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 2112 Vcvv 3398 Or wor 5452 infcinf 9035 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rmo 3059 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-po 5453 df-so 5454 df-cnv 5544 df-sup 9036 df-inf 9037 |
This theorem is referenced by: limsupval 15000 lcmval 16112 odzval 16307 ramval 16524 imasdsfn 16973 imasdsval 16974 odval 18880 odf 18883 gexval 18921 nmoval 23567 metdsval 23698 ovolval 24324 ovolf 24333 elqaalem1 25166 elqaalem3 25168 ballotlemi 32133 pellfundval 40346 dgraaval 40613 dgraaf 40616 liminfgval 42921 liminfval2 42927 ovnval2 43701 |
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