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Theorem untsucf 32954
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 1916 . . 3 𝑦 ¬ 𝑥𝑥
31, 2nfralw 3219 . 2 𝑦𝑥𝐴 ¬ 𝑥𝑥
4 vex 3482 . . . 4 𝑦 ∈ V
54elsuc 6241 . . 3 (𝑦 ∈ suc 𝐴 ↔ (𝑦𝐴𝑦 = 𝐴))
6 elequ1 2122 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑥))
7 elequ2 2130 . . . . . . 7 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
86, 7bitrd 282 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝑥𝑦𝑦))
98notbid 321 . . . . 5 (𝑥 = 𝑦 → (¬ 𝑥𝑥 ↔ ¬ 𝑦𝑦))
109rspccv 3605 . . . 4 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦𝐴 → ¬ 𝑦𝑦))
11 untelirr 32952 . . . . 5 (∀𝑥𝐴 ¬ 𝑥𝑥 → ¬ 𝐴𝐴)
12 eleq1 2903 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑦𝐴𝑦))
13 eleq2 2904 . . . . . . 7 (𝑦 = 𝐴 → (𝐴𝑦𝐴𝐴))
1412, 13bitrd 282 . . . . . 6 (𝑦 = 𝐴 → (𝑦𝑦𝐴𝐴))
1514notbid 321 . . . . 5 (𝑦 = 𝐴 → (¬ 𝑦𝑦 ↔ ¬ 𝐴𝐴))
1611, 15syl5ibrcom 250 . . . 4 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦 = 𝐴 → ¬ 𝑦𝑦))
1710, 16jaod 856 . . 3 (∀𝑥𝐴 ¬ 𝑥𝑥 → ((𝑦𝐴𝑦 = 𝐴) → ¬ 𝑦𝑦))
185, 17syl5bi 245 . 2 (∀𝑥𝐴 ¬ 𝑥𝑥 → (𝑦 ∈ suc 𝐴 → ¬ 𝑦𝑦))
193, 18ralrimi 3210 1 (∀𝑥𝐴 ¬ 𝑥𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦𝑦)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 844   = wceq 1538  wcel 2115  wnfc 2962  wral 3132  suc csuc 6174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-v 3481  df-un 3923  df-sn 4549  df-suc 6178
This theorem is referenced by:  dfon2lem3  33048
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