Theorem infex2g 7189

Description: Existence of infimum. (Contributed by Jim Kingdon, 1-Oct-2024.)

Assertion

Ref	Expression
infex2g	⊢ (𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)

Proof of Theorem infex2g

Step	Hyp	Ref	Expression
1		df-inf 7140	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		supex2g 7188	. 2 ⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, ◡𝑅) ∈ V)
3	1, 2	eqeltrid 2316	1 ⊢ (𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2200  Vcvv 2799  ^◡ccnv 4715  supcsup 7137  infcinf 7138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-un 4521

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-rab 2517  df-v 2801  df-in 3203  df-ss 3210  df-uni 3888  df-sup 7139  df-inf 7140

This theorem is referenced by:  odzval  12750