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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 8232 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
2 | infexd.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | cnvso 5591 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
4 | 2, 3 | sylib 207 | . . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
5 | 4 | supexd 8242 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ V) |
6 | 1, 5 | syl5eqel 2692 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 Vcvv 3173 Or wor 4958 ◡ccnv 5037 supcsup 8229 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: infex 8282 omsfval 29683 wsucex 31019 prmdvdsfmtnof1 40037 |
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