Theorem moeq 3013

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 2864	. . . 4 ⊢ (A ∈ V ↔ ∃x x = A)
2		eueq 3009	. . . 4 ⊢ (A ∈ V ↔ ∃!x x = A)
3	1, 2	bitr3i 242	. . 3 ⊢ (∃x x = A ↔ ∃!x x = A)
4	3	biimpi 186	. 2 ⊢ (∃x x = A → ∃!x x = A)
5		df-mo 2209	. 2 ⊢ (∃*x x = A ↔ (∃x x = A → ∃!x x = A))
6	4, 5	mpbir 200	1 ⊢ ∃*x x = A

Syntax hints:   → wi 4  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  ∃*wmo 2205  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  mosub  3015  euxfr2  3022  reueq  3034  funopabeq  5141  funsn  5148  fvopab4g  5389  ov2ag  5599  ov3  5600  ov6g  5601  ovmpt4g  5711  ovmpt2x  5713  fnce  6177