Theorem moeq 2982

Description: There is at most one set equal to a class. (Contributed by NM, 8-Mar-1995.)

Assertion

Ref	Expression
moeq	⊢ ∃*𝑥 𝑥 = 𝐴

Distinct variable group:   𝑥,𝐴

Proof of Theorem moeq

Step	Hyp	Ref	Expression
1		isset 2810	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		eueq 2978	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	bitr3i 186	. . 3 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃!𝑥 𝑥 = 𝐴)
4	3	biimpi 120	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴)
5		df-mo 2083	. 2 ⊢ (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴))
6	4, 5	mpbir 146	1 ⊢ ∃*𝑥 𝑥 = 𝐴

Syntax hints:   → wi 4   = wceq 1398  ∃wex 1541  ∃!weu 2079  ∃*wmo 2080   ∈ wcel 2202  Vcvv 2803

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2805

This theorem is referenced by:  euxfr2dc  2992  reueq  3006  mosn  3709  sndisj  4089  disjxsn  4091  reusv1  4561  funopabeq  5369  funcnvsn  5382  fvmptg  5731  fvopab6  5752  ovmpt4g  6154  ovi3  6169  ov6g  6170  oprabex3  6300  1stconst  6395  2ndconst  6396  axaddf  8148  axmulf  8149