Theorem predfrirr 6333

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 5653	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)
2		elpredg 6312	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
3	2	anidms 568	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
4	3	notbid 318	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
5	1, 4	imbitrrid 245	. . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝑅 Fr 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
6	5	expd 417	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑅 Fr 𝐴 → (𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))))
7	6	pm2.43b 55	. 2 ⊢ (𝑅 Fr 𝐴 → (𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
8		predel 6319	. . 3 ⊢ (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑋 ∈ 𝐴)
9	8	con3i 154	. 2 ⊢ (¬ 𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
10	7, 9	pm2.61d1 180	1 ⊢ (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∈ wcel 2107   class class class wbr 5148   Fr wfr 5628  Predcpred 6297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-fr 5631  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298

This theorem is referenced by:  frrlem12  8279  wfrlem14OLD  8319