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| Description: An infimum belongs to its base class (closure law). See also inflb 9530 and infglb 9531. (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 9484 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
| 2 | infcl.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
| 3 | cnvso 6307 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
| 4 | 2, 3 | sylib 218 | . . 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 9528 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) | 
| 7 | 4, 6 | supcl 9499 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) | 
| 8 | 1, 7 | eqeltrid 2844 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 class class class wbr 5142 Or wor 5590 ◡ccnv 5683 supcsup 9481 infcinf 9482 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-po 5591 df-so 5592 df-cnv 5692 df-iota 6513 df-riota 7389 df-sup 9483 df-inf 9484 | 
| This theorem is referenced by: infrecl 12251 infxrcl 13376 infssd 32723 xrge0infssd 32766 infxrge0lb 32769 infxrge0gelb 32771 omsf 34299 wzel 35826 wsuccl 35829 | 
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