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Theorem iinvdif 5073
Description: The indexed intersection of a complement. (Contributed by Gérard Lang, 5-Aug-2018.)
Assertion
Ref Expression
iinvdif 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem iinvdif
StepHypRef Expression
1 dif0 4364 . . . 4 (V ∖ ∅) = V
2 0iun 5056 . . . . 5 𝑥 ∈ ∅ 𝐵 = ∅
32difeq2i 4111 . . . 4 (V ∖ 𝑥 ∈ ∅ 𝐵) = (V ∖ ∅)
4 0iin 5057 . . . 4 𝑥 ∈ ∅ (V ∖ 𝐵) = V
51, 3, 43eqtr4ri 2763 . . 3 𝑥 ∈ ∅ (V ∖ 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵)
6 iineq1 5004 . . 3 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = 𝑥 ∈ ∅ (V ∖ 𝐵))
7 iuneq1 5003 . . . 4 (𝐴 = ∅ → 𝑥𝐴 𝐵 = 𝑥 ∈ ∅ 𝐵)
87difeq2d 4114 . . 3 (𝐴 = ∅ → (V ∖ 𝑥𝐴 𝐵) = (V ∖ 𝑥 ∈ ∅ 𝐵))
95, 6, 83eqtr4a 2790 . 2 (𝐴 = ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
10 iindif2 5070 . 2 (𝐴 ≠ ∅ → 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵))
119, 10pm2.61ine 3017 1 𝑥𝐴 (V ∖ 𝐵) = (V ∖ 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3466  cdif 3937  c0 4314   ciun 4987   ciin 4988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-in 3947  df-ss 3957  df-nul 4315  df-iun 4989  df-iin 4990
This theorem is referenced by: (None)
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