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Theorem compneOLD 39420
Description: Obsolete proof of compne 39419 as of 11-Nov-2021. (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
compneOLD (V ∖ 𝐴) ≠ 𝐴

Proof of Theorem compneOLD
StepHypRef Expression
1 vn0 4126 . 2 V ≠ ∅
2 ssun1 3975 . . . . . . . 8 V ⊆ (V ∪ 𝐴)
3 ssv 3822 . . . . . . . 8 (V ∪ 𝐴) ⊆ V
42, 3eqssi 3815 . . . . . . 7 V = (V ∪ 𝐴)
5 undif1 4238 . . . . . . 7 ((V ∖ 𝐴) ∪ 𝐴) = (V ∪ 𝐴)
64, 5eqtr4i 2825 . . . . . 6 V = ((V ∖ 𝐴) ∪ 𝐴)
7 uneq1 3959 . . . . . 6 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∪ 𝐴) = (𝐴𝐴))
86, 7syl5eq 2846 . . . . 5 ((V ∖ 𝐴) = 𝐴 → V = (𝐴𝐴))
9 unidm 3955 . . . . 5 (𝐴𝐴) = 𝐴
108, 9syl6eq 2850 . . . 4 ((V ∖ 𝐴) = 𝐴 → V = 𝐴)
11 difabs 4093 . . . . . . 7 ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
12 id 22 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
1311, 12syl5req 2847 . . . . . 6 ((V ∖ 𝐴) = 𝐴𝐴 = ((V ∖ 𝐴) ∖ 𝐴))
14 difeq1 3920 . . . . . 6 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴𝐴))
1513, 14eqtrd 2834 . . . . 5 ((V ∖ 𝐴) = 𝐴𝐴 = (𝐴𝐴))
16 difid 4150 . . . . 5 (𝐴𝐴) = ∅
1715, 16syl6eq 2850 . . . 4 ((V ∖ 𝐴) = 𝐴𝐴 = ∅)
1810, 17eqtrd 2834 . . 3 ((V ∖ 𝐴) = 𝐴 → V = ∅)
1918necon3i 3004 . 2 (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
201, 19ax-mp 5 1 (V ∖ 𝐴) ≠ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wne 2972  Vcvv 3386  cdif 3767  cun 3768  c0 4116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rab 3099  df-v 3388  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117
This theorem is referenced by: (None)
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