Theorem moeq 2913

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 2744	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		eueq 2909	. . . 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 2030	. 2 ⊢ (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴))
6	4, 5	mpbir 146	1 ⊢ ∃*𝑥 𝑥 = 𝐴

Syntax hints:   → wi 4   = wceq 1353  ∃wex 1492  ∃!weu 2026  ∃*wmo 2027   ∈ wcel 2148  Vcvv 2738

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-v 2740

This theorem is referenced by:  euxfr2dc  2923  reueq  2937  mosn  3629  sndisj  4000  disjxsn  4002  reusv1  4459  funopabeq  5253  funcnvsn  5262  fvmptg  5593  fvopab6  5613  ovmpt4g  5997  ovi3  6011  ov6g  6012  oprabex3  6130  1stconst  6222  2ndconst  6223  axaddf  7867  axmulf  7868