Theorem infex2g 7232

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 7183	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		supex2g 7231	. 2 ⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, ◡𝑅) ∈ V)
3	1, 2	eqeltrid 2318	1 ⊢ (𝐴 ∈ 𝐶 → inf(𝐵, 𝐴, 𝑅) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2202  Vcvv 2802  ^◡ccnv 4724  supcsup 7180  infcinf 7181

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-rab 2519  df-v 2804  df-in 3206  df-ss 3213  df-uni 3894  df-sup 7182  df-inf 7183

This theorem is referenced by:  odzval  12813