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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 8395 and infglb 8396. (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 8349 | . 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
2 | infcl.1 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
3 | cnvso 5674 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
4 | 2, 3 | sylib 208 | . . 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 8393 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
7 | 4, 6 | supcl 8364 | . 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
8 | 1, 7 | syl5eqel 2705 | 1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 class class class wbr 4653 Or wor 5034 ◡ccnv 5113 supcsup 8346 infcinf 8347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-po 5035 df-so 5036 df-cnv 5122 df-iota 5851 df-riota 6611 df-sup 8348 df-inf 8349 |
This theorem is referenced by: infrecl 11005 infxrcl 12163 infssd 29488 xrge0infssd 29526 infxrge0lb 29529 infxrge0gelb 29531 omsf 30358 wzel 31771 wsuccl 31776 |
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