Theorem supex2g 7010

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 6961	. 2 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		rabexg 4132	. . 3 ⊢ (𝐴 ∈ 𝐶 → {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
3	2	uniexd 4425	. 2 ⊢ (𝐴 ∈ 𝐶 → ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
4	1, 3	eqeltrid 2257	1 ⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∈ wcel 2141  ∀wral 2448  ∃wrex 2449  {crab 2452  Vcvv 2730  ∪ cuni 3796   class class class wbr 3989  supcsup 6959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-sup 6961

This theorem is referenced by:  infex2g  7011  pczpre  12251