Theorem predfrirr 6317

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 5621	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)
2		elpredg 6298	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
3	2	anidms 574	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
4	3	notbid 320	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
5	1, 4	imbitrrid 248	. . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝑅 Fr 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
6	5	expd 419	. . 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 6304	. . 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 181	1 ⊢ (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2141   class class class wbr 5099   Fr wfr 5595  Predcpred 6283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-fr 5598  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284

This theorem is referenced by:  frrlem12  8273