Theorem elsymdif 3224

Description: Membership in symmetric difference. (Contributed by SF, 10-Jan-2015.)

Assertion

Ref	Expression
elsymdif	⊢ (A ∈ (B ⊕ C) ↔ ¬ (A ∈ B ↔ A ∈ C))

Proof of Theorem elsymdif

Step	Hyp	Ref	Expression
1		elun 3221	. . 3 ⊢ (A ∈ ((B ∖ C) ∪ (C ∖ B)) ↔ (A ∈ (B ∖ C) ∨ A ∈ (C ∖ B)))
2		eldif 3222	. . . 4 ⊢ (A ∈ (B ∖ C) ↔ (A ∈ B ∧ ¬ A ∈ C))
3		eldif 3222	. . . 4 ⊢ (A ∈ (C ∖ B) ↔ (A ∈ C ∧ ¬ A ∈ B))
4	2, 3	orbi12i 507	. . 3 ⊢ ((A ∈ (B ∖ C) ∨ A ∈ (C ∖ B)) ↔ ((A ∈ B ∧ ¬ A ∈ C) ∨ (A ∈ C ∧ ¬ A ∈ B)))
5	1, 4	bitri 240	. 2 ⊢ (A ∈ ((B ∖ C) ∪ (C ∖ B)) ↔ ((A ∈ B ∧ ¬ A ∈ C) ∨ (A ∈ C ∧ ¬ A ∈ B)))
6		df-symdif 3217	. . 3 ⊢ (B ⊕ C) = ((B ∖ C) ∪ (C ∖ B))
7	6	eleq2i 2417	. 2 ⊢ (A ∈ (B ⊕ C) ↔ A ∈ ((B ∖ C) ∪ (C ∖ B)))
8		xor 861	. 2 ⊢ (¬ (A ∈ B ↔ A ∈ C) ↔ ((A ∈ B ∧ ¬ A ∈ C) ∨ (A ∈ C ∧ ¬ A ∈ B)))
9	5, 7, 8	3bitr4i 268	1 ⊢ (A ∈ (B ⊕ C) ↔ ¬ (A ∈ B ↔ A ∈ C))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∈ wcel 1710   ∖ cdif 3207   ∪ cun 3208   ⊕ csymdif 3210

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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217

This theorem is referenced by:  opkelimagekg  4272  dfaddc2  4382  nnsucelrlem1  4425  ltfinex  4465  eqpwrelk  4479  eqpw1relk  4480  eqtfinrelk  4487  evenfinex  4504  oddfinex  4505  evenodddisjlem1  4516  nnadjoinlem1  4520  srelk  4525  tfinnnlem1  4534  dfop2lem1  4574  setconslem2  4733  setconslem3  4734  setconslem7  4738  dfswap2  4742  brimage  5794  releqel  5808  releqmpt2  5810  extex  5916  qsexg  5983  ovcelem1  6172  tcfnex  6245