Theorem unisucg 4554

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 Jim Kingdon, 18-Aug-2019.)

Assertion

Ref	Expression
unisucg	⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))

Proof of Theorem unisucg

Step	Hyp	Ref	Expression
1		df-tr 4225	. . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
2		ssequn1 3399	. . 3 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
3	1, 2	bitri 184	. 2 ⊢ (Tr 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
4		df-suc 4511	. . . . . 6 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
5	4	unieqi 3940	. . . . 5 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
6		uniun 3949	. . . . 5 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
7	5, 6	eqtri 2259	. . . 4 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ ∪ {𝐴})
8		unisng 3947	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
9	8	uneq2d 3383	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
10	7, 9	eqtrid 2283	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))
11	10	eqeq1d 2247	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴))
12	3, 11	bitr4id 199	1 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ∪ cun 3218   ⊆ wss 3220  {csn 3705  ∪ cuni 3930  Tr wtr 4224  suc csuc 4505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-suc 4511

This theorem is referenced by:  onsucuni2  4706  nlimsucg  4708  ctmlemr  7438  nnnninfeq2  7459  nnsf  16953  peano4nninf  16954  nnnninfex  16970