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Theorem iundif2ss 3936
Description: Indexed union of class difference. Compare to theorem "De Morgan's laws" in [Enderton] p. 31. (Contributed by Jim Kingdon, 17-Aug-2018.)
Assertion
Ref Expression
iundif2ss 𝑥𝐴 (𝐵𝐶) ⊆ (𝐵 𝑥𝐴 𝐶)
Distinct variable group:   𝑥,𝐵
Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iundif2ss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eldif 3130 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
21rexbii 2477 . . . . 5 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
3 r19.42v 2627 . . . . 5 (∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
42, 3bitri 183 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
5 rexnalim 2459 . . . . . 6 (∃𝑥𝐴 ¬ 𝑦𝐶 → ¬ ∀𝑥𝐴 𝑦𝐶)
6 vex 2733 . . . . . . 7 𝑦 ∈ V
7 eliin 3876 . . . . . . 7 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
86, 7ax-mp 5 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
95, 8sylnibr 672 . . . . 5 (∃𝑥𝐴 ¬ 𝑦𝐶 → ¬ 𝑦 𝑥𝐴 𝐶)
109anim2i 340 . . . 4 ((𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶) → (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
114, 10sylbi 120 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) → (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
12 eliun 3875 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
13 eldif 3130 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
1411, 12, 133imtr4i 200 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) → 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
1514ssriv 3151 1 𝑥𝐴 (𝐵𝐶) ⊆ (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wcel 2141  wral 2448  wrex 2449  Vcvv 2730  cdif 3118  wss 3121   ciun 3871   ciin 3872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-iun 3873  df-iin 3874
This theorem is referenced by: (None)
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