Theorem predfrirr 5709

Description: Given a well-founded relationship, 𝑋 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 5091	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋𝑅𝑋)
2		elpredg 5694	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
3	2	anidms 677	. . . . . 6 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ 𝑋𝑅𝑋))
4	3	notbid 308	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ ¬ 𝑋𝑅𝑋))
5	1, 4	syl5ibr 236	. . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝑅 Fr 𝐴 ∧ 𝑋 ∈ 𝐴) → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋)))
6	5	expd 452	. . 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 5697	. . 3 ⊢ (𝑋 ∈ Pred(𝑅, 𝐴, 𝑋) → 𝑋 ∈ 𝐴)
9	8	con3i 150	. 2 ⊢ (¬ 𝑋 ∈ 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))
10	7, 9	pm2.61d1 171	1 ⊢ (𝑅 Fr 𝐴 → ¬ 𝑋 ∈ Pred(𝑅, 𝐴, 𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∈ wcel 1990   class class class wbr 4653   Fr wfr 5070  Predcpred 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-fr 5073  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680

This theorem is referenced by:  wfrlem14  7428