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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 8272 | . 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 1977 Vcvv 3173 Or wor 4958 infcinf 8230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rmo 2904 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-po 4959 df-so 4960 df-cnv 5046 df-sup 8231 df-inf 8232 |
This theorem is referenced by: limsupval 14053 lcmval 15143 odzval 15334 ramval 15550 imasdsfn 15997 imasdsval 15998 odval 17776 odf 17779 gexval 17816 nmoval 22329 metdsval 22458 ovolval 23049 ovolf 23057 elqaalem1 23878 elqaalem3 23880 ballotlemi 29889 pellfundval 36462 dgraaval 36733 dgraaf 36736 ovnval2 39435 |
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