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Theorem compneOLD 38961
Description: Obsolete proof of compne 38960 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 3957 . 2 V ≠ ∅
2 ssun1 3809 . . . . . . . 8 V ⊆ (V ∪ 𝐴)
3 ssv 3658 . . . . . . . 8 (V ∪ 𝐴) ⊆ V
42, 3eqssi 3652 . . . . . . 7 V = (V ∪ 𝐴)
5 undif1 4076 . . . . . . 7 ((V ∖ 𝐴) ∪ 𝐴) = (V ∪ 𝐴)
64, 5eqtr4i 2676 . . . . . 6 V = ((V ∖ 𝐴) ∪ 𝐴)
7 uneq1 3793 . . . . . 6 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∪ 𝐴) = (𝐴𝐴))
86, 7syl5eq 2697 . . . . 5 ((V ∖ 𝐴) = 𝐴 → V = (𝐴𝐴))
9 unidm 3789 . . . . 5 (𝐴𝐴) = 𝐴
108, 9syl6eq 2701 . . . 4 ((V ∖ 𝐴) = 𝐴 → V = 𝐴)
11 difabs 3925 . . . . . . 7 ((V ∖ 𝐴) ∖ 𝐴) = (V ∖ 𝐴)
12 id 22 . . . . . . 7 ((V ∖ 𝐴) = 𝐴 → (V ∖ 𝐴) = 𝐴)
1311, 12syl5req 2698 . . . . . 6 ((V ∖ 𝐴) = 𝐴𝐴 = ((V ∖ 𝐴) ∖ 𝐴))
14 difeq1 3754 . . . . . 6 ((V ∖ 𝐴) = 𝐴 → ((V ∖ 𝐴) ∖ 𝐴) = (𝐴𝐴))
1513, 14eqtrd 2685 . . . . 5 ((V ∖ 𝐴) = 𝐴𝐴 = (𝐴𝐴))
16 difid 3981 . . . . 5 (𝐴𝐴) = ∅
1715, 16syl6eq 2701 . . . 4 ((V ∖ 𝐴) = 𝐴𝐴 = ∅)
1810, 17eqtrd 2685 . . 3 ((V ∖ 𝐴) = 𝐴 → V = ∅)
1918necon3i 2855 . 2 (V ≠ ∅ → (V ∖ 𝐴) ≠ 𝐴)
201, 19ax-mp 5 1 (V ∖ 𝐴) ≠ 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wne 2823  Vcvv 3231  cdif 3604  cun 3605  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949
This theorem is referenced by: (None)
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