Theorem uncom 3367

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 736	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴))
2		elun 3364	. . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐴) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴))
3	1, 2	bitr4i 187	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐵 ∪ 𝐴))
4	3	uneqri 3365	1 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)

Syntax hints:   ∨ wo 716   = wceq 1398   ∈ wcel 2205   ∪ cun 3212

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218

This theorem is referenced by:  equncom  3368  uneq2  3371  un12  3381  un23  3382  ssun2  3387  unss2  3394  ssequn2  3396  undir  3475  dif32  3488  undif2ss  3589  uneqdifeqim  3599  prcom  3772  tpass  3792  prprc1  3805  difsnss  3845  exmid1stab  4326  suc0  4537  fununfun  5404  fresaunres2disj  5550  fresaunres1disj  5551  fvun2  5749  fmptpr  5881  fvsnun2  5887  fsnunfv  5890  omv2  6711  phplem2  7120  undifdc  7197  endjusym  7400  fzsuc2  10435  fseq1p1m1  10450  xnn0nnen  10823  hashfibclem  11231  ennnfonelem1  13242  setsslid  13347  lgsquadlem2  16063