Theorem unisuc 4116

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

Step	Hyp	Ref	Expression
1		ssequn1 3107	. 2 ⊢ (∪ A ⊆ A ↔ (∪ A ∪ A) = A)
2		df-tr 3846	. 2 ⊢ (Tr A ↔ ∪ A ⊆ A)
3		df-suc 4074	. . . . 5 ⊢ suc A = (A ∪ {A})
4	3	unieqi 3581	. . . 4 ⊢ ∪ suc A = ∪ (A ∪ {A})
5		uniun 3590	. . . 4 ⊢ ∪ (A ∪ {A}) = (∪ A ∪ ∪ {A})
6		unisuc.1	. . . . . 6 ⊢ A ∈ V
7	6	unisn 3587	. . . . 5 ⊢ ∪ {A} = A
8	7	uneq2i 3088	. . . 4 ⊢ (∪ A ∪ ∪ {A}) = (∪ A ∪ A)
9	4, 5, 8	3eqtri 2061	. . 3 ⊢ ∪ suc A = (∪ A ∪ A)
10	9	eqeq1i 2044	. 2 ⊢ (∪ suc A = A ↔ (∪ A ∪ A) = A)
11	1, 2, 10	3bitr4i 201	1 ⊢ (Tr A ↔ ∪ suc A = A)

Syntax hints:   ↔ wb 98   = wceq 1242   ∈ wcel 1390  Vcvv 2551   ∪ cun 2909   ⊆ wss 2911  {csn 3367  ∪ cuni 3571  Tr wtr 3845  suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-suc 4074

This theorem is referenced by:  onunisuci  4135  ordsucunielexmid  4216  tfrexlem  5889