Theorem predpoirr 6328

Description: Given a partial ordering, a class is not a member of its predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.)

Assertion

Ref	Expression
predpoirr	⊢ (𝑅 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

Proof of Theorem predpoirr

Step	Hyp	Ref	Expression
1		poirr 5593	. . . . 5 ⊢ ((𝑅 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)
2		elpredg 6308	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
3	2	anidms 566	. . . . . 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 Po 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
6	5	expd 415	. . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑅 Po 𝐴 → (𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))))
7	6	pm2.43b 55	. 2 ⊢ (𝑅 Po 𝐴 → (𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
8		predel 6315	. . 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 Po 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2098   class class class wbr 5141   Po wpo 5579  Predcpred 6293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-po 5581  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294

This theorem is referenced by:  xpord2indlem  8133  xpord3inddlem  8140