Theorem untsucf 35650

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 1910	. . 3 ⊢ Ⅎ𝑦 ¬ 𝑥 ∈ 𝑥
3	1, 2	nfralw 3307	. 2 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥
4		vex 3481	. . . 4 ⊢ 𝑦 ∈ V
5	4	elsuc 6451	. . 3 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
6		elequ1 2111	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
7		elequ2 2119	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
8	6, 7	bitrd 279	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
9	8	notbid 318	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
10	9	rspccv 3619	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ 𝐴 → ¬ 𝑦 ∈ 𝑦))
11		untelirr 35648	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)
12		eleq1 2825	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦))
13		eleq2 2826	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴))
14	12, 13	bitrd 279	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴))
15	14	notbid 318	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ 𝑦 ∈ 𝑦 ↔ ¬ 𝐴 ∈ 𝐴))
16	11, 15	syl5ibrcom 247	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 = 𝐴 → ¬ 𝑦 ∈ 𝑦))
17	10, 16	jaod 858	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → ¬ 𝑦 ∈ 𝑦))
18	5, 17	biimtrid 242	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ suc 𝐴 → ¬ 𝑦 ∈ 𝑦))
19	3, 18	ralrimi 3253	1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 846   = wceq 1535   ∈ wcel 2104  Ⅎwnfc 2886  ∀wral 3057  suc csuc 6383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1538  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-v 3479  df-un 3968  df-sn 4632  df-suc 6387

This theorem is referenced by:  dfon2lem3  35727