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Theorem iundif2ss 3967
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 3153 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
21rexbii 2497 . . . . 5 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
3 r19.42v 2647 . . . . 5 (∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
42, 3bitri 184 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
5 rexnalim 2479 . . . . . 6 (∃𝑥𝐴 ¬ 𝑦𝐶 → ¬ ∀𝑥𝐴 𝑦𝐶)
6 vex 2755 . . . . . . 7 𝑦 ∈ V
7 eliin 3906 . . . . . . 7 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
86, 7ax-mp 5 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
95, 8sylnibr 678 . . . . 5 (∃𝑥𝐴 ¬ 𝑦𝐶 → ¬ 𝑦 𝑥𝐴 𝐶)
109anim2i 342 . . . 4 ((𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶) → (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
114, 10sylbi 121 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) → (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
12 eliun 3905 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
13 eldif 3153 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
1411, 12, 133imtr4i 201 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) → 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
1514ssriv 3174 1 𝑥𝐴 (𝐵𝐶) ⊆ (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 104  wb 105  wcel 2160  wral 2468  wrex 2469  Vcvv 2752  cdif 3141  wss 3144   ciun 3901   ciin 3902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-iun 3903  df-iin 3904
This theorem is referenced by: (None)
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