Theorem moeq 2978

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 2806	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		eueq 2974	. . . 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 2081	. 2 ⊢ (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴))
6	4, 5	mpbir 146	1 ⊢ ∃*𝑥 𝑥 = 𝐴

Syntax hints:   → wi 4   = wceq 1395  ∃wex 1538  ∃!weu 2077  ∃*wmo 2078   ∈ wcel 2200  Vcvv 2799

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-v 2801

This theorem is referenced by:  euxfr2dc  2988  reueq  3002  mosn  3702  sndisj  4078  disjxsn  4080  reusv1  4548  funopabeq  5353  funcnvsn  5365  fvmptg  5709  fvopab6  5730  ovmpt4g  6126  ovi3  6141  ov6g  6142  oprabex3  6272  1stconst  6365  2ndconst  6366  axaddf  8051  axmulf  8052