Theorem infex2g 7276

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 7227	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		supex2g 7275	. 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 2803  ^◡ccnv 4730  supcsup 7224  infcinf 7225

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-rab 2520  df-v 2805  df-in 3207  df-ss 3214  df-uni 3899  df-sup 7226  df-inf 7227

This theorem is referenced by:  odzval  12877