Theorem un0 3278

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 3255	. . . 4 ⊢ ¬ 𝑥 ∈ ∅
2	1	biorfi 697	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅))
3	2	bicomi 130	. 2 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ∅) ↔ 𝑥 ∈ 𝐴)
4	3	uneqri 3114	1 ⊢ (𝐴 ∪ ∅) = 𝐴

Syntax hints:   ∨ wo 661   = wceq 1284   ∈ wcel 1433   ∪ cun 2971  ∅c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-nul 3252

This theorem is referenced by:  un00  3290  disjssun  3307  difun2  3322  difdifdirss  3327  disjpr2  3456  prprc1  3500  diftpsn3  3527  iununir  3759  suc0  4166  sucprc  4167  fvun1  5260  fmptpr  5376  fvunsng  5378  fvsnun1  5381  fvsnun2  5382  fsnunfv  5384  fsnunres  5385  rdg0  5997  omv2  6068  unsnfidcex  6385  fzsuc2  9096  fseq1p1m1  9111