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Theorem iundif2ss 3722
 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 2927 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
21rexbii 2331 . . . . 5 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
3 r19.42v 2467 . . . . 5 (∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
42, 3bitri 173 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
5 rexnalim 2317 . . . . . 6 (∃𝑥𝐴 ¬ 𝑦𝐶 → ¬ ∀𝑥𝐴 𝑦𝐶)
6 vex 2560 . . . . . . 7 𝑦 ∈ V
7 eliin 3662 . . . . . . 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 3661 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
13 eldif 2927 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
1411, 12, 133imtr4i 190 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) → 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
1514ssriv 2949 1 𝑥𝐴 (𝐵𝐶) ⊆ (𝐵 𝑥𝐴 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  Vcvv 2557   ∖ cdif 2914   ⊆ wss 2917  ∪ ciun 3657  ∩ ciin 3658 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 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-iun 3659  df-iin 3660 This theorem is referenced by: (None)
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