Theorem moeq 3382

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 3207	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
2		eueq 3378	. . 3 ⊢ (𝐴 ∈ V ↔ ∃!𝑥 𝑥 = 𝐴)
3	1, 2	sylbb1 227	. 2 ⊢ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴)
4		df-mo 2475	. 2 ⊢ (∃*𝑥 𝑥 = 𝐴 ↔ (∃𝑥 𝑥 = 𝐴 → ∃!𝑥 𝑥 = 𝐴))
5	3, 4	mpbir 221	1 ⊢ ∃*𝑥 𝑥 = 𝐴

Syntax hints:   → wi 4   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∃*wmo 2471  Vcvv 3200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  mosub  3384  euxfr2  3391  reueq  3404  rmoeq  3405  sndisj  4644  disjxsn  4646  reusv1OLD  4867  reuxfr2d  4891  funopabeq  5924  funcnvsn  5936  fvmptg  6280  fvopab6  6310  ovmpt4g  6783  ov3  6797  ov6g  6798  oprabex3  7157  1stconst  7265  2ndconst  7266  iunmapdisj  8846  axaddf  9966  axmulf  9967  joinfval  17001  joinval  17005  meetfval  17015  meetval  17019  reuxfr3d  29329  abrexdom2jm  29346  abrexdom2  33526