Theorem iunconst 4961

Description: Indexed union of a constant class, i.e. where 𝐵 does not depend on 𝑥. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iunconst

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4955	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
2		r19.9rzv 4461	. . 3 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
3	1, 2	bitr4id 293	. 2 ⊢ (𝐴 ≠ ∅ → (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑦 ∈ 𝐵))
4	3	eqrdv 2759	1 ⊢ (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∃wrex 3087  ∅c0 4279  ∪ ciun 4951

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

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-v 3453  df-dif 3902  df-nul 4280  df-iun 4953

This theorem is used by:  iununi  5059  oe1m  8546  oarec  8563  oelim2  8597  scottrankd  9942  bnj1143  35413  poimirlem32  38550  mblfinlem2  38556  hoicvr  47527  ovnlecvr2  47589  iunhoiioo  47655