Theorem uncom 3409

Description: Commutative law for union of classes. Exercise 6 of [TakeutiZaring] p. 17. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
uncom	⊢ (A ∪ B) = (B ∪ A)

Proof of Theorem uncom

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orcom 376	. . 3 ⊢ ((x ∈ A ∨ x ∈ B) ↔ (x ∈ B ∨ x ∈ A))
2		elun 3221	. . 3 ⊢ (x ∈ (B ∪ A) ↔ (x ∈ B ∨ x ∈ A))
3	1, 2	bitr4i 243	. 2 ⊢ ((x ∈ A ∨ x ∈ B) ↔ x ∈ (B ∪ A))
4	3	uneqri 3407	1 ⊢ (A ∪ B) = (B ∪ A)

Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ∪ cun 3208

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-un 3215

This theorem is referenced by:  equncom  3410  uneq2  3413  un12  3422  un23  3423  ssun2  3428  unss2  3435  ssequn2  3437  undir  3505  unineq  3506  dif32  3518  symdifcom  3543  disjpss  3602  undif1  3626  undif2  3627  difcom  3635  uneqdifeq  3639  dfif4  3674  dfif5  3675  prcom  3799  tpass  3819  difsnid  3855  ssunsn2  3866  sspr  3870  sstp  3871  pwadjoin  4120  addccom  4407  sfinltfin  4536  phialllem2  4618  fvun2  5381  fvsnun2  5449  sbthlem1  6204