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

Theorem unisuc 6289
Description: A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
unisuc.1 𝐴 ∈ V
Assertion
Ref Expression
unisuc (Tr 𝐴 suc 𝐴 = 𝐴)

Proof of Theorem unisuc
StepHypRef Expression
1 ssequn1 4094 . 2 ( 𝐴𝐴 ↔ ( 𝐴𝐴) = 𝐴)
2 df-tr 5162 . 2 (Tr 𝐴 𝐴𝐴)
3 df-suc 6219 . . . . 5 suc 𝐴 = (𝐴 ∪ {𝐴})
43unieqi 4832 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
5 uniun 4844 . . . 4 (𝐴 ∪ {𝐴}) = ( 𝐴 {𝐴})
6 unisuc.1 . . . . . 6 𝐴 ∈ V
76unisn 4841 . . . . 5 {𝐴} = 𝐴
87uneq2i 4074 . . . 4 ( 𝐴 {𝐴}) = ( 𝐴𝐴)
94, 5, 83eqtri 2769 . . 3 suc 𝐴 = ( 𝐴𝐴)
109eqeq1i 2742 . 2 ( suc 𝐴 = 𝐴 ↔ ( 𝐴𝐴) = 𝐴)
111, 2, 103bitr4i 306 1 (Tr 𝐴 suc 𝐴 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1543  wcel 2110  Vcvv 3408  cun 3864  wss 3866  {csn 4541   cuni 4819  Tr wtr 5161  suc csuc 6215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-un 3871  df-in 3873  df-ss 3883  df-sn 4542  df-pr 4544  df-uni 4820  df-tr 5162  df-suc 6219
This theorem is referenced by:  onunisuci  6327  ordunisuc  7611
  Copyright terms: Public domain W3C validator