Theorem dmsnn0 5065

Description: The domain of a singleton is nonzero iff the singleton argument is a set. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Scott Fenton, 19-Apr-2021.)

Assertion

Ref	Expression
dmsnn0	⊢ (A ∈ V ↔ dom {A} ≠ ∅)

Proof of Theorem dmsnn0

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldm2 4900	. . . 4 ⊢ (x ∈ dom {A} ↔ ∃y⟨x, y⟩ ∈ {A})
2		vex 2863	. . . . . . . 8 ⊢ x ∈ V
3		vex 2863	. . . . . . . 8 ⊢ y ∈ V
4	2, 3	opex 4589	. . . . . . 7 ⊢ ⟨x, y⟩ ∈ V
5	4	elsnc 3757	. . . . . 6 ⊢ (⟨x, y⟩ ∈ {A} ↔ ⟨x, y⟩ = A)
6		eqcom 2355	. . . . . 6 ⊢ (⟨x, y⟩ = A ↔ A = ⟨x, y⟩)
7	5, 6	bitri 240	. . . . 5 ⊢ (⟨x, y⟩ ∈ {A} ↔ A = ⟨x, y⟩)
8	7	exbii 1582	. . . 4 ⊢ (∃y⟨x, y⟩ ∈ {A} ↔ ∃y A = ⟨x, y⟩)
9	1, 8	bitri 240	. . 3 ⊢ (x ∈ dom {A} ↔ ∃y A = ⟨x, y⟩)
10	9	exbii 1582	. 2 ⊢ (∃x x ∈ dom {A} ↔ ∃x∃y A = ⟨x, y⟩)
11		n0 3560	. 2 ⊢ (dom {A} ≠ ∅ ↔ ∃x x ∈ dom {A})
12		opeqexb 4621	. 2 ⊢ (A ∈ V ↔ ∃x∃y A = ⟨x, y⟩)
13	10, 11, 12	3bitr4ri 269	1 ⊢ (A ∈ V ↔ dom {A} ≠ ∅)

Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710   ≠ wne 2517  Vcvv 2860  ∅c0 3551  {csn 3738  ⟨cop 4562  dom cdm 4773

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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-cnv 4786  df-rn 4787  df-dm 4788

This theorem is referenced by:  rnsnn0  5066