Theorem adj11 3890

Description: Adjoining a new element is one-to-one. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
adj11	⊢ ((¬ C ∈ A ∧ ¬ C ∈ B) → ((A ∪ {C}) = (B ∪ {C}) ↔ A = B))

Proof of Theorem adj11

Step	Hyp	Ref	Expression
1		difeq1 3247	. . . 4 ⊢ ((A ∪ {C}) = (B ∪ {C}) → ((A ∪ {C}) ∖ {C}) = ((B ∪ {C}) ∖ {C}))
2		difun2 3630	. . . 4 ⊢ ((A ∪ {C}) ∖ {C}) = (A ∖ {C})
3		difun2 3630	. . . 4 ⊢ ((B ∪ {C}) ∖ {C}) = (B ∖ {C})
4	1, 2, 3	3eqtr3g 2408	. . 3 ⊢ ((A ∪ {C}) = (B ∪ {C}) → (A ∖ {C}) = (B ∖ {C}))
5		difsn 3846	. . . 4 ⊢ (¬ C ∈ A → (A ∖ {C}) = A)
6		difsn 3846	. . . 4 ⊢ (¬ C ∈ B → (B ∖ {C}) = B)
7	5, 6	eqeqan12d 2368	. . 3 ⊢ ((¬ C ∈ A ∧ ¬ C ∈ B) → ((A ∖ {C}) = (B ∖ {C}) ↔ A = B))
8	4, 7	syl5ib 210	. 2 ⊢ ((¬ C ∈ A ∧ ¬ C ∈ B) → ((A ∪ {C}) = (B ∪ {C}) → A = B))
9		uneq1 3412	. 2 ⊢ (A = B → (A ∪ {C}) = (B ∪ {C}))
10	8, 9	impbid1 194	1 ⊢ ((¬ C ∈ A ∧ ¬ C ∈ B) → ((A ∪ {C}) = (B ∪ {C}) ↔ A = B))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207   ∪ cun 3208  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742

This theorem is referenced by:  nnadjoin  4521