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Theorem difunieq 36559
Description: The difference of unions is a subset of the union of the difference. (Contributed by ML, 29-Mar-2021.)
Assertion
Ref Expression
difunieq ( 𝐴 𝐵) ⊆ (𝐴𝐵)

Proof of Theorem difunieq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 4912 . . . 4 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
2 eluni 4912 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
32notbii 319 . . . 4 𝑥 𝐵 ↔ ¬ ∃𝑦(𝑥𝑦𝑦𝐵))
4 alinexa 1844 . . . . . 6 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) ↔ ¬ ∃𝑦(𝑥𝑦𝑦𝐵))
5 nfa1 2147 . . . . . . 7 𝑦𝑦(𝑥𝑦 → ¬ 𝑦𝐵)
6 sp 2175 . . . . . . . . . 10 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → (𝑥𝑦 → ¬ 𝑦𝐵))
76adantrd 491 . . . . . . . . 9 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → ((𝑥𝑦𝑦𝐴) → ¬ 𝑦𝐵))
87ancld 550 . . . . . . . 8 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → ((𝑥𝑦𝑦𝐴) → ((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦𝐵)))
9 anass 468 . . . . . . . 8 (((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦𝐵) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
108, 9imbitrdi 250 . . . . . . 7 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → ((𝑥𝑦𝑦𝐴) → (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵))))
115, 10eximd 2208 . . . . . 6 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → (∃𝑦(𝑥𝑦𝑦𝐴) → ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵))))
124, 11sylbir 234 . . . . 5 (¬ ∃𝑦(𝑥𝑦𝑦𝐵) → (∃𝑦(𝑥𝑦𝑦𝐴) → ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵))))
1312impcom 407 . . . 4 ((∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ ∃𝑦(𝑥𝑦𝑦𝐵)) → ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
141, 3, 13syl2anb 597 . . 3 ((𝑥 𝐴 ∧ ¬ 𝑥 𝐵) → ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
15 eldif 3959 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴 ∧ ¬ 𝑥 𝐵))
16 eluni 4912 . . . 4 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
17 eldif 3959 . . . . . 6 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴 ∧ ¬ 𝑦𝐵))
1817anbi2i 622 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
1918exbii 1849 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
2016, 19bitri 274 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
2114, 15, 203imtr4i 291 . 2 (𝑥 ∈ ( 𝐴 𝐵) → 𝑥 (𝐴𝐵))
2221ssriv 3987 1 ( 𝐴 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1538  wex 1780  wcel 2105  cdif 3946  wss 3949   cuni 4909
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-10 2136  ax-12 2170  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3952  df-in 3956  df-ss 3966  df-uni 4910
This theorem is referenced by:  inunissunidif  36560
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