Theorem supex2g 7099

Description: Existence of supremum. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
supex2g	⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)

Proof of Theorem supex2g

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sup 7050	. 2 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		rabexg 4176	. . 3 ⊢ (𝐴 ∈ 𝐶 → {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
3	2	uniexd 4475	. 2 ⊢ (𝐴 ∈ 𝐶 → ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
4	1, 3	eqeltrid 2283	1 ⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  {crab 2479  Vcvv 2763  ∪ cuni 3839   class class class wbr 4033  supcsup 7048

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-rab 2484  df-v 2765  df-in 3163  df-ss 3170  df-uni 3840  df-sup 7050

This theorem is referenced by:  infex2g  7100  pczpre  12466