Theorem un0 3525

Description: The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
un0	⊢ (𝐴 ∪ ∅) = 𝐴

Proof of Theorem un0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3495	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
2	1	biorfi 751	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅))
3	2	bicomi 132	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅) ↔ 𝑥 ∈ 𝐴)
4	3	uneqri 3346	1 ⊢ (𝐴 ∪ ∅) = 𝐴

Syntax hints:   ∨ wo 713   = wceq 1395   ∈ wcel 2200   ∪ cun 3195  ∅c0 3491

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-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492

This theorem is referenced by:  un00  3538  disjssun  3555  difun2  3571  difdifdirss  3576  disjpr2  3730  prprc1  3775  diftpsn3  3809  iununir  4049  exmid1stab  4292  suc0  4502  sucprc  4503  fvun1  5702  fmptpr  5835  fvunsng  5837  fvsnun1  5840  fvsnun2  5841  fsnunfv  5844  fsnunres  5845  rdg0  6539  omv2  6619  unsnfidcex  7090  unfidisj  7092  undifdc  7094  ssfirab  7106  dju0en  7404  djuassen  7407  fmelpw1o  7440  fzsuc2  10283  fseq1p1m1  10298  hashunlem  11034  ennnfonelem1  12986  setsresg  13078  setsslid  13091  lgsquadlem2  15765