Theorem uneqri 4099

Description: Inference from membership to union. (Contributed by NM, 21-Jun-1993.)

Hypothesis

Ref	Expression
uneqri.1	⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)

Assertion

Ref	Expression
uneqri	⊢ (𝐴 ∪ 𝐵) = 𝐶

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem uneqri

Step	Hyp	Ref	Expression
1		elun 4096	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		uneqri.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)
3	1, 2	bitri 274	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ 𝐶)
4	3	eqriv 2733	1 ⊢ (𝐴 ∪ 𝐵) = 𝐶

Syntax hints:   ↔ wb 205   ∨ wo 844   = wceq 1540   ∈ wcel 2105   ∪ cun 3896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3443  df-un 3903

This theorem is referenced by:  unidm  4100  uncom  4101  unass  4114  dfun2  4207  undi  4222  unab  4246  un0  4338  inundif  4426