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Theorem disjxun0 30332
Description: Simplify a disjoint union. (Contributed by Thierry Arnoux, 27-Nov-2023.)
Hypothesis
Ref Expression
disjxun0.1 ((𝜑𝑥𝐵) → 𝐶 = ∅)
Assertion
Ref Expression
disjxun0 (𝜑 → (Disj 𝑥 ∈ (𝐴𝐵)𝐶Disj 𝑥𝐴 𝐶))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem disjxun0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 disjxun0.1 . . . . 5 ((𝜑𝑥𝐵) → 𝐶 = ∅)
2 nel02 4270 . . . . 5 (𝐶 = ∅ → ¬ 𝑦𝐶)
31, 2syl 17 . . . 4 ((𝜑𝑥𝐵) → ¬ 𝑦𝐶)
43rmounid 30264 . . 3 (𝜑 → (∃*𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ ∃*𝑥𝐴 𝑦𝐶))
54albidv 1921 . 2 (𝜑 → (∀𝑦∃*𝑥 ∈ (𝐴𝐵)𝑦𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶))
6 df-disj 5008 . 2 (Disj 𝑥 ∈ (𝐴𝐵)𝐶 ↔ ∀𝑦∃*𝑥 ∈ (𝐴𝐵)𝑦𝐶)
7 df-disj 5008 . 2 (Disj 𝑥𝐴 𝐶 ↔ ∀𝑦∃*𝑥𝐴 𝑦𝐶)
85, 6, 73bitr4g 317 1 (𝜑 → (Disj 𝑥 ∈ (𝐴𝐵)𝐶Disj 𝑥𝐴 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wcel 2114  ∃*wrmo 3133  cun 3906  c0 4265  Disj wdisj 5007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-mo 2622  df-clab 2801  df-cleq 2815  df-clel 2894  df-rmo 3138  df-v 3471  df-dif 3911  df-un 3913  df-nul 4266  df-disj 5008
This theorem is referenced by:  tocyccntz  30817
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