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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 9440 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
2 | infexd.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | cnvso 6281 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
4 | 2, 3 | sylib 217 | . . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
5 | 4 | supexd 9450 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ V) |
6 | 1, 5 | eqeltrid 2831 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 Vcvv 3468 Or wor 5580 ◡ccnv 5668 supcsup 9437 infcinf 9438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-po 5581 df-so 5582 df-cnv 5677 df-sup 9439 df-inf 9440 |
This theorem is referenced by: infex 9490 omsfval 33823 wsucex 35331 prproropf1olem4 46751 prmdvdsfmtnof1 46832 |
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