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Mirrors > Home > MPE Home > Th. List > infexd | Structured version Visualization version GIF version |
Description: An infimum is a set. (Contributed by AV, 2-Sep-2020.) |
Ref | Expression |
---|---|
infexd.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
Ref | Expression |
---|---|
infexd | ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 8637 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
2 | infexd.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | cnvso 5928 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
4 | 2, 3 | sylib 210 | . . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
5 | 4 | supexd 8647 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ V) |
6 | 1, 5 | syl5eqel 2863 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3398 Or wor 5273 ◡ccnv 5354 supcsup 8634 infcinf 8635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rmo 3098 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-po 5274 df-so 5275 df-cnv 5363 df-sup 8636 df-inf 8637 |
This theorem is referenced by: infex 8687 omsfval 30954 wsucex 32360 prproropf1olem4 42445 prmdvdsfmtnof1 42520 |
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