Theorem supex2g 7147

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 7098	. 2 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		rabexg 4192	. . 3 ⊢ (𝐴 ∈ 𝐶 → {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
3	2	uniexd 4492	. 2 ⊢ (𝐴 ∈ 𝐶 → ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
4	1, 3	eqeltrid 2293	1 ⊢ (𝐴 ∈ 𝐶 → sup(𝐵, 𝐴, 𝑅) ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∈ wcel 2177  ∀wral 2485  ∃wrex 2486  {crab 2489  Vcvv 2773  ∪ cuni 3853   class class class wbr 4048  supcsup 7096

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-un 4485

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-rab 2494  df-v 2775  df-in 3174  df-ss 3181  df-uni 3854  df-sup 7098

This theorem is referenced by:  infex2g  7148  pczpre  12670