Theorem infcl 8936

Description: An infimum belongs to its base class (closure law). See also inflb 8937 and infglb 8938. (Contributed by AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infcl.1	⊢ (𝜑 → 𝑅 Or 𝐴)
infcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))

Assertion

Ref	Expression
infcl	⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem infcl

Step	Hyp	Ref	Expression
1		df-inf 8891	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		infcl.1	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
3		cnvso 6107	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
4	2, 3	sylib 221	. . 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 8935	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
7	4, 6	supcl 8906	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
8	1, 7	eqeltrid 2894	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   class class class wbr 5030   Or wor 5437  ^◡ccnv 5518  supcsup 8888  infcinf 8889

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-po 5438  df-so 5439  df-cnv 5527  df-iota 6283  df-riota 7093  df-sup 8890  df-inf 8891

This theorem is referenced by:  infrecl  11610  infxrcl  12714  infssd  30472  xrge0infssd  30511  infxrge0lb  30514  infxrge0gelb  30516  omsf  31664  wzel  33224  wsuccl  33227