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Theorem conss34OLD 39065
Description: Obsolete proof of complss 3859 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 39060 . . . . 5 (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥𝐵)
3 compel 39060 . . . . 5 (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥𝐴)
42, 3imbi12i 339 . . . 4 ((𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)) ↔ (¬ 𝑥𝐵 → ¬ 𝑥𝐴))
51, 4bitr4i 267 . . 3 ((𝑥𝐴𝑥𝐵) ↔ (𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
65albii 1860 . 2 (∀𝑥(𝑥𝐴𝑥𝐵) ↔ ∀𝑥(𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
7 dfss2 3697 . 2 (𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
8 dfss2 3697 . 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 1594  wcel 2103  Vcvv 3304  cdif 3677  wss 3680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-v 3306  df-dif 3683  df-in 3687  df-ss 3694
This theorem is referenced by: (None)
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