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Theorem conss34OLD 38114
Description: Obsolete proof of complss 3734 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 306 . . . 4 ((𝑥𝐴𝑥𝐵) ↔ (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
2 compel 38109 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
3 compel 38109 . . . . 5 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
42, 3imbi12i 340 . . . 4 ((𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)) ↔ (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
51, 4bitr4i 267 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
65albii 1744 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
7 dfss2 3577 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
8 dfss2 3577 . 2 ((V ∖ 𝐵) ⊆ (V ∖ 𝐴) ↔ ∀𝑥(𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
96, 7, 83bitr4i 292 1 (𝐴𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478  wcel 1992  Vcvv 3191  cdif 3557  wss 3560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-dif 3563  df-in 3567  df-ss 3574
This theorem is referenced by: (None)
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