Theorem moeq 3001

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 2828	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		eueq 2997	. . . 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 2090	. 2 ⊢ (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴))
6	4, 5	mpbir 146	1 ⊢ ∃*𝑥 𝑥 = 𝐴

This proof depends on syntax axioms:   → wi 4   = wceq 1402  ∃wex 1545  ∃!weu 2086  ∃*wmo 2087   ∈ wcel 2209  Vcvv 2821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823

This theorem is used by:  euxfr2dc  3011  reueq  3025  mosn  3745  sndisj  4126  disjxsn  4128  reusv1  4604  funopabeq  5413  funcnvsn  5426  fvmptg  5781  fvopab6  5805  ovmpt4g  6211  ovi3  6226  ov6g  6227  oprabex3  6362  1stconst  6457  2ndconst  6458  axaddf  8235  axmulf  8236