Theorem untsucf 36057

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 1934	. . 3 ⊢ Ⅎ𝑦 ¬ 𝑥 ∈ 𝑥
3	1, 2	nfralw 3309	. 2 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥
4		vex 3458	. . . 4 ⊢ 𝑦 ∈ V
5	4	elsuc 6418	. . 3 ⊢ (𝑦 ∈ suc 𝐴 ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴))
6		elequ1 2149	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥))
7		elequ2 2157	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
8	6, 7	bitrd 281	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑥 ↔ 𝑦 ∈ 𝑦))
9	8	notbid 320	. . . . 5 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ 𝑥 ↔ ¬ 𝑦 ∈ 𝑦))
10	9	rspccv 3578	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ 𝐴 → ¬ 𝑦 ∈ 𝑦))
11		untelirr 36055	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)
12		eleq1 2850	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦))
13		eleq2 2851	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴))
14	12, 13	bitrd 281	. . . . . 6 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑦 ↔ 𝐴 ∈ 𝐴))
15	14	notbid 320	. . . . 5 ⊢ (𝑦 = 𝐴 → (¬ 𝑦 ∈ 𝑦 ↔ ¬ 𝐴 ∈ 𝐴))
16	11, 15	syl5ibrcom 249	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 = 𝐴 → ¬ 𝑦 ∈ 𝑦))
17	10, 16	jaod 870	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ((𝑦 ∈ 𝐴 ∨ 𝑦 = 𝐴) → ¬ 𝑦 ∈ 𝑦))
18	5, 17	biimtrid 244	. 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → (𝑦 ∈ suc 𝐴 → ¬ 𝑦 ∈ 𝑦))
19	3, 18	ralrimi 3260	1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 858   = wceq 1560   ∈ wcel 2142  Ⅎwnfc 2909  ∀wral 3076  suc csuc 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-v 3456  df-un 3909  df-sn 4583  df-suc 6352

This theorem is referenced by:  dfon2lem3  36130