Theorem uncom 3081

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 646	. . 3 ⊢ ((x ∈ A ∨ x ∈ B) ↔ (x ∈ B ∨ x ∈ A))
2		elun 3078	. . 3 ⊢ (x ∈ (B ∪ A) ↔ (x ∈ B ∨ x ∈ A))
3	1, 2	bitr4i 176	. 2 ⊢ ((x ∈ A ∨ x ∈ B) ↔ x ∈ (B ∪ A))
4	3	uneqri 3079	1 ⊢ (A ∪ B) = (B ∪ A)

Syntax hints:   ∨ wo 628   = wceq 1242   ∈ wcel 1390   ∪ cun 2909

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916

This theorem is referenced by:  equncom  3082  uneq2  3085  un12  3095  un23  3096  ssun2  3101  unss2  3108  ssequn2  3110  undir  3181  dif32  3194  disjpss  3272  undif2ss  3293  uneqdifeqim  3302  prcom  3437  tpass  3457  prprc1  3469  difsnss  3501  suc0  4114  fvun2  5183  fmptpr  5298  fvsnun2  5304  fsnunfv  5306  omv2  5984  fzsuc2  8711  fseq1p1m1  8726