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Mirrors > Home > MPE Home > Th. List > infcl | Structured version Visualization version GIF version |
Description: An infimum belongs to its base class (closure law). See also inflb 9480 and infglb 9481. (Contributed by AV, 3-Sep-2020.) |
Ref | Expression |
---|---|
infcl.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
infcl.2 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) |
Ref | Expression |
---|---|
infcl | ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 9434 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
2 | infcl.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | cnvso 6284 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
4 | 2, 3 | sylib 217 | . . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
5 | infcl.2 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
6 | 2, 5 | infcllem 9478 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
7 | 4, 6 | supcl 9449 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
8 | 1, 7 | eqeltrid 2837 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 class class class wbr 5147 Or wor 5586 ◡ccnv 5674 supcsup 9431 infcinf 9432 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-po 5587 df-so 5588 df-cnv 5683 df-iota 6492 df-riota 7361 df-sup 9433 df-inf 9434 |
This theorem is referenced by: infrecl 12192 infxrcl 13308 infssd 31922 xrge0infssd 31961 infxrge0lb 31964 infxrge0gelb 31966 omsf 33283 wzel 34784 wsuccl 34787 |
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