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Theorem iundif2ss 3719
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 2924 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
21rexbii 2328 . . . . 5 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
3 r19.42v 2464 . . . . 5 (∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
42, 3bitri 173 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
5 rexnalim 2314 . . . . . 6 (∃𝑥𝐴 ¬ 𝑦𝐶 → ¬ ∀𝑥𝐴 𝑦𝐶)
6 vex 2557 . . . . . . 7 𝑦 ∈ V
7 eliin 3659 . . . . . . 7 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
86, 7ax-mp 7 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
95, 8sylnibr 602 . . . . 5 (∃𝑥𝐴 ¬ 𝑦𝐶 → ¬ 𝑦 𝑥𝐴 𝐶)
109anim2i 324 . . . 4 ((𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶) → (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
114, 10sylbi 114 . . 3 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) → (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
12 eliun 3658 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
13 eldif 2924 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
1411, 12, 133imtr4i 190 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) → 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
1514ssriv 2946 1 𝑥𝐴 (𝐵𝐶) ⊆ (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 97  wb 98  wcel 1393  wral 2303  wrex 2304  Vcvv 2554  cdif 2911  wss 2914   ciun 3654   ciin 3655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-dif 2917  df-in 2921  df-ss 2928  df-iun 3656  df-iin 3657
This theorem is referenced by: (None)
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