Theorem eueq 2427

Description: Equality has existential uniqueness.

Assertion

Ref	Expression
eueq	|- (A e. _V <-> E!x x = A)

Distinct variable group:   x,A

Proof of Theorem eueq

Step	Hyp	Ref	Expression
1		eqtr3 1907	. . . 4 |- ((x = A /\ y = A) -> x = y)
2	1	gen2 1329	. . 3 |- A.xA.y((x = A /\ y = A) -> x = y)
3	2	biantru 793	. 2 |- (E.x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
4		isset 2296	. 2 |- (A e. _V <-> E.x x = A)
5		eqeq1 1890	. . 3 |- (x = y -> (x = A <-> y = A))
6	5	eu4 1806	. 2 |- (E!x x = A <-> (E.x x = A /\ A.xA.y((x = A /\ y = A) -> x = y)))
7	3, 4, 6	3bitr4i 200	1 |- (A e. _V <-> E!x x = A)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  _Vcvv 2292

This theorem is referenced by:  eueq1 2428  moeq 2431  0exOLD 3447  snexOLD 3493  euuni 3807  reuhypd 3848  fnopab2g 4547  fvopab2 4754  elrnopabg 4773  fopab2 4796  en2d 5459  hartog 5693  upxp 10225  fopab2g 14485  hartogOLD 15384  upixp 15729  iotasbc 16383

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294

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