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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 8278 and infglb 8279. (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 8232 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
2 | infcl.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 | infcl.2 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) | |
6 | 2, 5 | infcllem 8276 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
7 | 4, 6 | supcl 8247 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
8 | 1, 7 | syl5eqel 2692 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 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-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 |
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-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 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-iota 5768 df-riota 6511 df-sup 8231 df-inf 8232 |
This theorem is referenced by: infrecl 10882 infxrcl 12035 infssd 28871 xrge0infssd 28916 infxrge0lb 28919 infxrge0gelb 28921 omsf 29685 wzel 31015 wsuccl 31020 |
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