Theorem untsucf 33651

Description: If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypothesis

Ref	Expression
untsucf.1	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
untsucf	⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem untsucf

Step	Hyp	Ref	Expression
1		untsucf.1	. . 3 ⊢ Ⅎ𝑦𝐴
2		nfv 1917	. . 3 ⊢ Ⅎ𝑦 ¬ 𝑥 ∈ 𝑥
3	1, 2	nfralw 3151	. 2 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥
4		vex 3436	. . . 4 ⊢ 𝑦 ∈ V
5	4	elsuc 6335	. . 3 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
6		elequ1 2113	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
7		elequ2 2121	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
8	6, 7	bitrd 278	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
9	8	notbid 318	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
10	9	rspccv 3558	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ 𝐴 → ¬ 𝑦 ∈ 𝑦))
11		untelirr 33649	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)
12		eleq1 2826	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦))
13		eleq2 2827	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴))
14	12, 13	bitrd 278	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴))
15	14	notbid 318	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ 𝑦 ∈ 𝑦 ↔ ¬ 𝐴 ∈ 𝐴))
16	11, 15	syl5ibrcom 246	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 = 𝐴 → ¬ 𝑦 ∈ 𝑦))
17	10, 16	jaod 856	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → ¬ 𝑦 ∈ 𝑦))
18	5, 17	syl5bi 241	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ suc 𝐴 → ¬ 𝑦 ∈ 𝑦))
19	3, 18	ralrimi 3141	1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Ⅎwnfc 2887  ∀wral 3064  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-v 3434  df-un 3892  df-sn 4562  df-suc 6272

This theorem is referenced by:  dfon2lem3  33761