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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 9375 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
2 | infexd.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | cnvso 6238 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
4 | 2, 3 | sylib 217 | . . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
5 | 4 | supexd 9385 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ V) |
6 | 1, 5 | eqeltrid 2842 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 Vcvv 3443 Or wor 5542 ◡ccnv 5630 supcsup 9372 infcinf 9373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-po 5543 df-so 5544 df-cnv 5639 df-sup 9374 df-inf 9375 |
This theorem is referenced by: infex 9425 omsfval 32763 wsucex 34271 prproropf1olem4 45630 prmdvdsfmtnof1 45711 |
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