Theorem iunconst 2000

Description: Indexed union of a constant class, i.e. where B does not depend on x.

Assertion

Ref	Expression
iunconst	⊢ (¬ A = ∅ → ∪x ∈ A B = B)

Distinct variable group(s):   x,A   x,B

Proof of Theorem iunconst

Step	Hyp	Ref	Expression
1		n0 1714	. . . 4 ⊢ (¬ A = ∅ ↔ ∃x x ∈ A)
2		ibar 487	. . . 4 ⊢ (∃x x ∈ A → (y ∈ B ↔ (∃x x ∈ A ∧ y ∈ B)))
3	1, 2	sylbi 174	. . 3 ⊢ (¬ A = ∅ → (y ∈ B ↔ (∃x x ∈ A ∧ y ∈ B)))
4		eliun 1998	. . . 4 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B)
5		df-rex 1206	. . . 4 ⊢ (∃x ∈ A y ∈ B ↔ ∃x(x ∈ A ∧ y ∈ B))
6		19.41v 963	. . . 4 ⊢ (∃x(x ∈ A ∧ y ∈ B) ↔ (∃x x ∈ A ∧ y ∈ B))
7	4, 5, 6	3bitr 155	. . 3 ⊢ (y ∈ ∪x ∈ A B ↔ (∃x x ∈ A ∧ y ∈ B))
8	3, 7	syl6rbbr 417	. 2 ⊢ (¬ A = ∅ → (y ∈ ∪x ∈ A B ↔ y ∈ B))
9	8	cleqrd 1100	1 ⊢ (¬ A = ∅ → ∪x ∈ A B = B)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ∅c0 1707  ∪ciun 1994

This theorem is referenced by:  abianfplem 2999  oe1m 3147

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-nul 1708  df-iun 1996

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