Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  extid Structured version   Visualization version   GIF version

Theorem extid 35012
Description: Property of identity relation, see also extep 34984, extssr 35194 and the comment of df-ssr 35183. (Contributed by Peter Mazsa, 5-Jul-2019.)
Assertion
Ref Expression
extid (𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))

Proof of Theorem extid
StepHypRef Expression
1 cnvi 5834 . . . . 5 I = I
21eceq2i 8124 . . . 4 [𝐴] I = [𝐴] I
3 ecidsn 8136 . . . 4 [𝐴] I = {𝐴}
42, 3eqtri 2796 . . 3 [𝐴] I = {𝐴}
51eceq2i 8124 . . . 4 [𝐵] I = [𝐵] I
6 ecidsn 8136 . . . 4 [𝐵] I = {𝐵}
75, 6eqtri 2796 . . 3 [𝐵] I = {𝐵}
84, 7eqeq12i 2786 . 2 ([𝐴] I = [𝐵] I ↔ {𝐴} = {𝐵})
9 sneqbg 4642 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
108, 9syl5bb 275 1 (𝐴𝑉 → ([𝐴] I = [𝐵] I ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1507  wcel 2050  {csn 4435   I cid 5305  ccnv 5400  [cec 8081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5054  ax-nul 5061  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-br 4924  df-opab 4986  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-ec 8085
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator