Theorem un0 3576

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 3555	. . . 4 ⊢ ¬ x ∈ ∅
2	1	biorfi 396	. . 3 ⊢ (x ∈ A ↔ (x ∈ A ∨ x ∈ ∅))
3	2	bicomi 193	. 2 ⊢ ((x ∈ A ∨ x ∈ ∅) ↔ x ∈ A)
4	3	uneqri 3407	1 ⊢ (A ∪ ∅) = A

Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ∪ cun 3208  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-nul 3552

This theorem is referenced by:  un00  3587  disjssun  3609  difun2  3630  difdifdir  3638  diftpsn3  3850  sspr  3870  sstp  3871  iununi  4051  prprc2  4123  addcid1  4406  nnsucelrlem3  4427  fvun1  5380  fvunsn  5445  fvsnun1  5448  fvsnun2  5449  sbthlem1  6204