Theorem uncom 3362

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 3359	. . 3 ⊢ (𝑥 ∈ (𝐵 ∪ 𝐴) ↔ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐴))
3	1, 2	bitr4i 187	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ (𝐵 ∪ 𝐴))
4	3	uneqri 3360	1 ⊢ (𝐴 ∪ 𝐵) = (𝐵 ∪ 𝐴)

Syntax hints:   ∨ wo 716   = wceq 1398   ∈ wcel 2203   ∪ cun 3208

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-un 3214

This theorem is referenced by:  equncom  3363  uneq2  3366  un12  3376  un23  3377  ssun2  3382  unss2  3389  ssequn2  3391  undir  3470  dif32  3483  undif2ss  3584  uneqdifeqim  3594  prcom  3766  tpass  3786  prprc1  3799  difsnss  3839  exmid1stab  4320  suc0  4531  fununfun  5398  fresaunres2disj  5544  fresaunres1disj  5545  fvun2  5743  fmptpr  5875  fvsnun2  5881  fsnunfv  5884  omv2  6697  phplem2  7106  undifdc  7183  endjusym  7386  fzsuc2  10412  fseq1p1m1  10427  xnn0nnen  10798  hashfibclem  11202  ennnfonelem1  13150  setsslid  13255  lgsquadlem2  15943