Theorem elsymdif 3223

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 3220	. . 3 ⊢ (A ∈ ((B ∖ C) ∪ (C ∖ B)) ↔ (A ∈ (B ∖ C) ∨ A ∈ (C ∖ B)))
2		eldif 3221	. . . 4 ⊢ (A ∈ (B ∖ C) ↔ (A ∈ B ∧ ¬ A ∈ C))
3		eldif 3221	. . . 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 3216	. . 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 3206   ∪ cun 3207   ⊕ csymdif 3209

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 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216

This theorem is referenced by:  opkelimagekg  4271  dfaddc2  4381  nnsucelrlem1  4424  ltfinex  4464  eqpwrelk  4478  eqpw1relk  4479  eqtfinrelk  4486  evenfinex  4503  oddfinex  4504  evenodddisjlem1  4515  nnadjoinlem1  4519  srelk  4524  tfinnnlem1  4533  dfop2lem1  4573  setconslem2  4732  setconslem3  4733  setconslem7  4737  dfswap2  4741  brimage  5793  releqel  5807  releqmpt2  5809  extex  5915  qsexg  5982  ovcelem1  6171  tcfnex  6244