Theorem difeqri2 10402

Description: Inference from membership to difference.

Assertion

Ref	Expression
difeqri2	⊢ (∀x((x ∈ A ⋀ ¬ x ∈ B) ↔ x ∈ C) → (A ∖ B) = C)

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem difeqri2

Step	Hyp	Ref	Expression
1		bicom 519	. . . . 5 ⊢ (((x ∈ A ⋀ ¬ x ∈ B) ↔ x ∈ C) ↔ (x ∈ C ↔ (x ∈ A ⋀ ¬ x ∈ B)))
2	1	albii 998	. . . 4 ⊢ (∀x((x ∈ A ⋀ ¬ x ∈ B) ↔ x ∈ C) ↔ ∀x(x ∈ C ↔ (x ∈ A ⋀ ¬ x ∈ B)))
3	2	biimp 151	. . 3 ⊢ (∀x((x ∈ A ⋀ ¬ x ∈ B) ↔ x ∈ C) → ∀x(x ∈ C ↔ (x ∈ A ⋀ ¬ x ∈ B)))
4		abeq2 1566	. . 3 ⊢ (C = {x∣(x ∈ A ⋀ ¬ x ∈ B)} ↔ ∀x(x ∈ C ↔ (x ∈ A ⋀ ¬ x ∈ B)))
5	3, 4	sylibr 200	. 2 ⊢ (∀x((x ∈ A ⋀ ¬ x ∈ B) ↔ x ∈ C) → C = {x∣(x ∈ A ⋀ ¬ x ∈ B)})
6		df-dif 2046	. 2 ⊢ (A ∖ B) = {x∣(x ∈ A ⋀ ¬ x ∈ B)}
7	5, 6	syl6reqr 1524	1 ⊢ (∀x((x ∈ A ⋀ ¬ x ∈ B) ↔ x ∈ C) → (A ∖ B) = C)

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 953   = wceq 955   ∈ wcel 957  {cab 1462   ∖ cdif 2041

This theorem is referenced by:  cdrci 10440

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-12 967  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-sb 1171  df-clab 1463  df-cleq 1468  df-clel 1471  df-dif 2046

Copyright terms: Public domain