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Theorem conss34OLD 37667
 Description: Obsolete proof of complss 3713 as of 7-Aug-2021. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
conss34OLD (𝐴𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))

Proof of Theorem conss34OLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 con34b 305 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
2 compel 37663 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
3 compel 37663 . . . . 5 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
42, 3imbi12i 339 . . . 4 ((𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)) ↔ (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
51, 4bitr4i 266 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
65albii 1737 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
7 dfss2 3557 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
8 dfss2 3557 . 2 ((V ∖ 𝐵) ⊆ (V ∖ 𝐴) ↔ ∀𝑥(𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
96, 7, 83bitr4i 291 1 (𝐴𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  ∀wal 1473   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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