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Theorem iundif2ss 3882
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 3081 . . . . . 6 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦𝐶))
21rexbii 2443 . . . . 5 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶))
3 r19.42v 2589 . . . . 5 (∃𝑥𝐴 (𝑦𝐵 ∧ ¬ 𝑦𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
42, 3bitri 183 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵 ∧ ∃𝑥𝐴 ¬ 𝑦𝐶))
5 rexnalim 2428 . . . . . 6 (∃𝑥𝐴 ¬ 𝑦𝐶 → ¬ ∀𝑥𝐴 𝑦𝐶)
6 vex 2690 . . . . . . 7 𝑦 ∈ V
7 eliin 3822 . . . . . . 7 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶))
86, 7ax-mp 5 . . . . . 6 (𝑦 𝑥𝐴 𝐶 ↔ ∀𝑥𝐴 𝑦𝐶)
95, 8sylnibr 667 . . . . 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 3821 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
13 eldif 3081 . . 3 (𝑦 ∈ (𝐵 𝑥𝐴 𝐶) ↔ (𝑦𝐵 ∧ ¬ 𝑦 𝑥𝐴 𝐶))
1411, 12, 133imtr4i 200 . 2 (𝑦 𝑥𝐴 (𝐵𝐶) → 𝑦 ∈ (𝐵 𝑥𝐴 𝐶))
1514ssriv 3102 1 𝑥𝐴 (𝐵𝐶) ⊆ (𝐵 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103  wb 104  wcel 1481  wral 2417  wrex 2418  Vcvv 2687  cdif 3069  wss 3072   ciun 3817   ciin 3818
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-dif 3074  df-in 3078  df-ss 3085  df-iun 3819  df-iin 3820
This theorem is referenced by: (None)
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