Theorem eluni1 4173

Description: Membership in a unit union. (Contributed by SF, 15-Mar-2015.)

Hypothesis

Ref	Expression
eluni1.1	⊢ A ∈ V

Assertion

Ref	Expression
eluni1	⊢ (A ∈ ⋃1B ↔ {A} ∈ B)

Proof of Theorem eluni1

Step	Hyp	Ref	Expression
1		eluni1.1	. 2 ⊢ A ∈ V
2		eluni1g 4172	. 2 ⊢ (A ∈ V → (A ∈ ⋃1B ↔ {A} ∈ B))
3	1, 2	ax-mp 5	1 ⊢ (A ∈ ⋃1B ↔ {A} ∈ B)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  Vcvv 2859  {csn 3737  ⋃₁cuni1 4133

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  ax-nin 4078  ax-sn 4087

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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892  df-1c 4136  df-uni1 4138

This theorem is referenced by:  dfuni12  4291  dfuni3  4315  dfint3  4318  ncfinraiselem2  4480  nnpweqlem1  4522  sfintfinlem1  4531  tfinnnlem1  4533  vfinspclt  4552  setconslem4  4734  enpw1lem1  6061  nenpw1pwlem1  6084  nmembers1lem1  6268  nchoicelem16  6304  nchoicelem18  6306