MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  euen1b Structured version   Visualization version   GIF version

Theorem euen1b 8257
Description: Two ways to express "𝐴 has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
euen1b (𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem euen1b
StepHypRef Expression
1 euen1 8256 . 2 (∃!𝑥 𝑥𝐴 ↔ {𝑥𝑥𝐴} ≈ 1𝑜)
2 abid2 2925 . . 3 {𝑥𝑥𝐴} = 𝐴
32breq1i 4844 . 2 ({𝑥𝑥𝐴} ≈ 1𝑜𝐴 ≈ 1𝑜)
41, 3bitr2i 267 1 (𝐴 ≈ 1𝑜 ↔ ∃!𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wcel 2155  ∃!weu 2629  {cab 2788   class class class wbr 4837  1𝑜c1o 7783  cen 8183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-br 4838  df-opab 4900  df-id 5213  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-1o 7790  df-en 8187
This theorem is referenced by:  euhash1  13419  f1otrspeq  18062  hausflf2  22009  minveclem4a  23407
  Copyright terms: Public domain W3C validator