Theorem djueq12 9864

Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)

Assertion

Ref	Expression
djueq12	⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷))

Proof of Theorem djueq12

Step	Hyp	Ref	Expression
1		xpeq2 5670	. . . 4 ⊢ (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵))
2	1	adantr 484	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵))
3		xpeq2 5670	. . . 4 ⊢ (𝐶 = 𝐷 → ({1o} × 𝐶) = ({1o} × 𝐷))
4	3	adantl 485	. . 3 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → ({1o} × 𝐶) = ({1o} × 𝐷))
5	2, 4	uneq12d 4124	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) = (({∅} × 𝐵) ∪ ({1o} × 𝐷)))
6		df-dju 9861	. 2 ⊢ (𝐴 ⊔ 𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶))
7		df-dju 9861	. 2 ⊢ (𝐵 ⊔ 𝐷) = (({∅} × 𝐵) ∪ ({1o} × 𝐷))
8	5, 6, 7	3eqtr4g 2824	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊔ 𝐶) = (𝐵 ⊔ 𝐷))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∪ cun 3904  ∅c0 4287  {csn 4584   × cxp 5647  1_oc1o 8432   ⊔ cdju 9858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-un 3911  df-opab 5165  df-xp 5655  df-dju 9861

This theorem is referenced by:  djueq1  9865  djueq2  9866  isfin5  10258  alephadd  10537