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Theorem untsucf 31713
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 1883 . . 3 𝑦 ¬ 𝑥𝑥
31, 2nfral 2974 . 2 𝑦𝑥𝐴 ¬ 𝑥𝑥
4 vex 3234 . . . 4 𝑦 ∈ V
54elsuc 5832 . . 3 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
6 elequ1 2037 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
7 elequ2 2044 . . . . . . 7 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
86, 7bitrd 268 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
98notbid 307 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
109rspccv 3337 . . . 4 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦𝐴 → ¬ 𝑦𝑦))
11 untelirr 31711 . . . . 5 (∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
12 eleq1 2718 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑦𝐴𝑦))
13 eleq2 2719 . . . . . . 7 (𝑦 = 𝐴 → (𝐴𝑦𝐴𝐴))
1412, 13bitrd 268 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑦𝐴𝐴))
1514notbid 307 . . . . 5 (𝑦 = 𝐴 → (¬ 𝑦𝑦 ↔ ¬ 𝐴𝐴))
1611, 15syl5ibrcom 237 . . . 4 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦 = 𝐴 → ¬ 𝑦𝑦))
1710, 16jaod 394 . . 3 (∀𝑥𝐴 ¬ 𝑥𝑥 → ((𝑦𝐴𝑦 = 𝐴) → ¬ 𝑦𝑦))
185, 17syl5bi 232 . 2 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦 ∈ suc 𝐴 → ¬ 𝑦𝑦))
193, 18ralrimi 2986 1 (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382   = wceq 1523  wcel 2030  wnfc 2780  wral 2941  suc csuc 5763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-un 3612  df-sn 4211  df-suc 5767
This theorem is referenced by:  dfon2lem3  31814
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