Theorem infexd 8933

Description: An infimum is a set. (Contributed by AV, 2-Sep-2020.)

Hypothesis

Ref	Expression
infexd.1	⊢ (𝜑 → 𝑅 Or 𝐴)

Assertion

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

Proof of Theorem infexd

Step	Hyp	Ref	Expression
1		df-inf 8893	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		infexd.1	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
3		cnvso 6125	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
4	2, 3	sylib 220	. . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴)
5	4	supexd 8903	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ V)
6	1, 5	eqeltrid 2917	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3486   Or wor 5459  ^◡ccnv 5540  supcsup 8890  infcinf 8891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pr 5316  ax-un 7447

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rmo 3146  df-rab 3147  df-v 3488  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-nul 4280  df-if 4454  df-sn 4554  df-pr 4556  df-op 4560  df-uni 4825  df-br 5053  df-opab 5115  df-po 5460  df-so 5461  df-cnv 5549  df-sup 8892  df-inf 8893

This theorem is referenced by:  infex  8943  omsfval  31559  wsucex  33120  prproropf1olem4  43753  prmdvdsfmtnof1  43834