Theorem untsucf 36475

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 1947	. . 3 ⊢ Ⅎ𝑦 ¬ 𝑥 ∈ 𝑥
3	1, 2	nfralw 3310	. 2 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥
4		vex 3455	. . . 4 ⊢ 𝑦 ∈ V
5	4	elsuc 6435	. . 3 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
6		elequ1 2152	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
7		elequ2 2160	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
8	6, 7	bitrd 282	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
9	8	notbid 321	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
10	9	rspccv 3574	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ 𝐴 → ¬ 𝑦 ∈ 𝑦))
11		untelirr 36473	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)
12		eleq1 2849	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦))
13		eleq2 2850	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴))
14	12, 13	bitrd 282	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴))
15	14	notbid 321	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ 𝑦 ∈ 𝑦 ↔ ¬ 𝐴 ∈ 𝐴))
16	11, 15	syl5ibrcom 250	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 = 𝐴 → ¬ 𝑦 ∈ 𝑦))
17	10, 16	jaod 873	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → ¬ 𝑦 ∈ 𝑦))
18	5, 17	biimtrid 245	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ suc 𝐴 → ¬ 𝑦 ∈ 𝑦))
19	3, 18	ralrimi 3261	1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2908  ∀wral 3077  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-v 3453  df-un 3904  df-sn 4585  df-suc 6368

This theorem is used by:  dfon2lem3  36547