Theorem conss34OLD 38646

Description: Obsolete proof of complss 3751 as of 7-Aug-2021. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
conss34OLD	⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))

Proof of Theorem conss34OLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		con34b 306	. . . 4 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))
2		compel 38641	. . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐵) ↔ ¬ 𝑥 ∈ 𝐵)
3		compel 38641	. . . . 5 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
4	2, 3	imbi12i 340	. . . 4 ⊢ ((𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)) ↔ (¬ 𝑥 ∈ 𝐵 → ¬ 𝑥 ∈ 𝐴))
5	1, 4	bitr4i 267	. . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
6	5	albii 1747	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
7		dfss2 3591	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
8		dfss2 3591	. 2 ⊢ ((V ∖ 𝐵) ⊆ (V ∖ 𝐴) ↔ ∀𝑥(𝑥 ∈ (V ∖ 𝐵) → 𝑥 ∈ (V ∖ 𝐴)))
9	6, 7, 8	3bitr4i 292	1 ⊢ (𝐴 ⊆ 𝐵 ↔ (V ∖ 𝐵) ⊆ (V ∖ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196  ∀wal 1481   ∈ wcel 1990  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588

This theorem is referenced by: (None)