Theorem supexd 9429

Description: A supremum is a set. (Contributed by NM, 22-May-1999.) (Revised by Mario Carneiro, 24-Dec-2016.)

Hypothesis

Ref	Expression
supmo.1	⊢ (𝜑 → 𝑅 Or 𝐴)

Assertion

Ref	Expression
supexd	⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)

Proof of Theorem supexd

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

Step	Hyp	Ref	Expression
1		df-sup 9418	. 2 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		supmo.1	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
3	2	supmo 9428	. . 3 ⊢ (𝜑 → ∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
4		rmorabex 5428	. . 3 ⊢ (∃*𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
5		uniexg 7746	. . 3 ⊢ ({𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V → ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
6	3, 4, 5	3syl 19	. 2 ⊢ (𝜑 → ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ∈ V)
7	1, 6	eqeltrid 2865	1 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ V)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  ∃*wrmo 3365  {crab 3413  Vcvv 3451  ∪ cuni 4867   class class class wbr 5103   Or wor 5558  supcsup 9416

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rmo 3366  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-po 5559  df-so 5560  df-sup 9418

This theorem is used by:  supex  9440  infexd  9460  smflimsuplem7  47780  prproropf1olem4  48532