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Theorem iundif2ss 3713
 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 x A (B𝐶) ⊆ (B x A 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 2921 . . . . . 6 (y (B𝐶) ↔ (y B ¬ y 𝐶))
21rexbii 2325 . . . . 5 (x A y (B𝐶) ↔ x A (y B ¬ y 𝐶))
3 r19.42v 2461 . . . . 5 (x A (y B ¬ y 𝐶) ↔ (y B x A ¬ y 𝐶))
42, 3bitri 173 . . . 4 (x A y (B𝐶) ↔ (y B x A ¬ y 𝐶))
5 rexnalim 2311 . . . . . 6 (x A ¬ y 𝐶 → ¬ x A y 𝐶)
6 vex 2554 . . . . . . 7 y V
7 eliin 3653 . . . . . . 7 (y V → (y x A 𝐶x A y 𝐶))
86, 7ax-mp 7 . . . . . 6 (y x A 𝐶x A y 𝐶)
95, 8sylnibr 601 . . . . 5 (x A ¬ y 𝐶 → ¬ y x A 𝐶)
109anim2i 324 . . . 4 ((y B x A ¬ y 𝐶) → (y B ¬ y x A 𝐶))
114, 10sylbi 114 . . 3 (x A y (B𝐶) → (y B ¬ y x A 𝐶))
12 eliun 3652 . . 3 (y x A (B𝐶) ↔ x A y (B𝐶))
13 eldif 2921 . . 3 (y (B x A 𝐶) ↔ (y B ¬ y x A 𝐶))
1411, 12, 133imtr4i 190 . 2 (y x A (B𝐶) → y (B x A 𝐶))
1514ssriv 2943 1 x A (B𝐶) ⊆ (B x A 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301  Vcvv 2551   ∖ cdif 2908   ⊆ wss 2911  ∪ ciun 3648  ∩ ciin 3649 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-in 2918  df-ss 2925  df-iun 3650  df-iin 3651 This theorem is referenced by: (None)
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