Theorem predfrirr 6336

Description: Given a well-founded relation, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 22-Apr-2011.)

Assertion

Ref	Expression
predfrirr	⊢ (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

Proof of Theorem predfrirr

Step	Hyp	Ref	Expression
1		frirr 5635	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)
2		elpredg 6317	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
3	2	anidms 577	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
4	3	notbid 321	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
5	1, 4	imbitrrid 249	. . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝑅 Fr 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
6	5	expd 421	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑅 Fr 𝐴 → (𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))))
7	6	pm2.43b 56	. 2 ⊢ (𝑅 Fr 𝐴 → (𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
8		predel 6323	. . 3 ⊢ (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑋 ∈ 𝐴)
9	8	con3i 155	. 2 ⊢ (¬ 𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
10	7, 9	pm2.61d1 182	1 ⊢ (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145   class class class wbr 5107   Fr wfr 5609  Predcpred 6302

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-fr 5612  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303

This theorem is used by:  frrlem12  8300