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Theorem difunieq 34752
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 4816 . . . 4 (𝑥 𝐴 ↔ ∃𝑦(𝑥𝑦𝑦𝐴))
2 eluni 4816 . . . . 5 (𝑥 𝐵 ↔ ∃𝑦(𝑥𝑦𝑦𝐵))
32notbii 323 . . . 4 𝑥 𝐵 ↔ ¬ ∃𝑦(𝑥𝑦𝑦𝐵))
4 alinexa 1844 . . . . . 6 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) ↔ ¬ ∃𝑦(𝑥𝑦𝑦𝐵))
5 nfa1 2155 . . . . . . 7 𝑦𝑦(𝑥𝑦 → ¬ 𝑦𝐵)
6 sp 2183 . . . . . . . . . 10 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → (𝑥𝑦 → ¬ 𝑦𝐵))
76adantrd 495 . . . . . . . . 9 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → ((𝑥𝑦𝑦𝐴) → ¬ 𝑦𝐵))
87ancld 554 . . . . . . . 8 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → ((𝑥𝑦𝑦𝐴) → ((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦𝐵)))
9 anass 472 . . . . . . . 8 (((𝑥𝑦𝑦𝐴) ∧ ¬ 𝑦𝐵) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
108, 9syl6ib 254 . . . . . . 7 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → ((𝑥𝑦𝑦𝐴) → (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵))))
115, 10eximd 2217 . . . . . 6 (∀𝑦(𝑥𝑦 → ¬ 𝑦𝐵) → (∃𝑦(𝑥𝑦𝑦𝐴) → ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵))))
124, 11sylbir 238 . . . . 5 (¬ ∃𝑦(𝑥𝑦𝑦𝐵) → (∃𝑦(𝑥𝑦𝑦𝐴) → ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵))))
1312impcom 411 . . . 4 ((∃𝑦(𝑥𝑦𝑦𝐴) ∧ ¬ ∃𝑦(𝑥𝑦𝑦𝐵)) → ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
141, 3, 13syl2anb 600 . . 3 ((𝑥 𝐴 ∧ ¬ 𝑥 𝐵) → ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
15 eldif 3918 . . 3 (𝑥 ∈ ( 𝐴 𝐵) ↔ (𝑥 𝐴 ∧ ¬ 𝑥 𝐵))
16 eluni 4816 . . . 4 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)))
17 eldif 3918 . . . . . 6 (𝑦 ∈ (𝐴𝐵) ↔ (𝑦𝐴 ∧ ¬ 𝑦𝐵))
1817anbi2i 625 . . . . 5 ((𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ (𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
1918exbii 1849 . . . 4 (∃𝑦(𝑥𝑦𝑦 ∈ (𝐴𝐵)) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
2016, 19bitri 278 . . 3 (𝑥 (𝐴𝐵) ↔ ∃𝑦(𝑥𝑦 ∧ (𝑦𝐴 ∧ ¬ 𝑦𝐵)))
2114, 15, 203imtr4i 295 . 2 (𝑥 ∈ ( 𝐴 𝐵) → 𝑥 (𝐴𝐵))
2221ssriv 3946 1 ( 𝐴 𝐵) ⊆ (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  wal 1536  wex 1781  wcel 2114  cdif 3905  wss 3908   cuni 4813
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-10 2145  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-dif 3911  df-in 3915  df-ss 3925  df-uni 4814
This theorem is referenced by:  inunissunidif  34753
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