Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  untsucf Structured version   Visualization version   GIF version

Theorem untsucf 35710
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 𝑦𝐴
Assertion
Ref Expression
untsucf (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦𝑦)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem untsucf
StepHypRef Expression
1 untsucf.1 . . 3 𝑦𝐴
2 nfv 1914 . . 3 𝑦 ¬ 𝑥𝑥
31, 2nfralw 3311 . 2 𝑦𝑥𝐴 ¬ 𝑥𝑥
4 vex 3484 . . . 4 𝑦 ∈ V
54elsuc 6454 . . 3 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
6 elequ1 2115 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
7 elequ2 2123 . . . . . . 7 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
86, 7bitrd 279 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
98notbid 318 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
109rspccv 3619 . . . 4 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦𝐴 → ¬ 𝑦𝑦))
11 untelirr 35708 . . . . 5 (∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
12 eleq1 2829 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑦𝐴𝑦))
13 eleq2 2830 . . . . . . 7 (𝑦 = 𝐴 → (𝐴𝑦𝐴𝐴))
1412, 13bitrd 279 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑦𝐴𝐴))
1514notbid 318 . . . . 5 (𝑦 = 𝐴 → (¬ 𝑦𝑦 ↔ ¬ 𝐴𝐴))
1611, 15syl5ibrcom 247 . . . 4 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦 = 𝐴 → ¬ 𝑦𝑦))
1710, 16jaod 860 . . 3 (∀𝑥𝐴 ¬ 𝑥𝑥 → ((𝑦𝐴𝑦 = 𝐴) → ¬ 𝑦𝑦))
185, 17biimtrid 242 . 2 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦 ∈ suc 𝐴 → ¬ 𝑦𝑦))
193, 18ralrimi 3257 1 (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 848   = wceq 1540  wcel 2108  wnfc 2890  wral 3061  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-v 3482  df-un 3956  df-sn 4627  df-suc 6390
This theorem is referenced by:  dfon2lem3  35786
  Copyright terms: Public domain W3C validator