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Theorem djueq12 9367
 Description: Equality theorem for disjoint union. (Contributed by Jim Kingdon, 23-Jun-2022.)
Assertion
Ref Expression
djueq12 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem djueq12
StepHypRef Expression
1 xpeq2 5546 . . . 4 (𝐴 = 𝐵 → ({∅} × 𝐴) = ({∅} × 𝐵))
21adantr 485 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({∅} × 𝐴) = ({∅} × 𝐵))
3 xpeq2 5546 . . . 4 (𝐶 = 𝐷 → ({1o} × 𝐶) = ({1o} × 𝐷))
43adantl 486 . . 3 ((𝐴 = 𝐵𝐶 = 𝐷) → ({1o} × 𝐶) = ({1o} × 𝐷))
52, 4uneq12d 4070 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (({∅} × 𝐴) ∪ ({1o} × 𝐶)) = (({∅} × 𝐵) ∪ ({1o} × 𝐷)))
6 df-dju 9364 . 2 (𝐴𝐶) = (({∅} × 𝐴) ∪ ({1o} × 𝐶))
7 df-dju 9364 . 2 (𝐵𝐷) = (({∅} × 𝐵) ∪ ({1o} × 𝐷))
85, 6, 73eqtr4g 2819 1 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∪ cun 3857  ∅c0 4226  {csn 4523   × cxp 5523  1oc1o 8106   ⊔ cdju 9361 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-un 3864  df-opab 5096  df-xp 5531  df-dju 9364 This theorem is referenced by:  djueq1  9368  djueq2  9369  isfin5  9760  alephadd  10038
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