Theorem wfis 6347

Description: Well-Ordered Induction Schema. If all elements less than a given set 𝑥 of the well-ordered class 𝐴 have a property (induction hypothesis), then all elements of 𝐴 have that property. (Contributed by Scott Fenton, 29-Jan-2011.)

Hypotheses

Ref	Expression
wfis.1	⊢ 𝑅 We 𝐴
wfis.2	⊢ 𝑅 Se 𝐴
wfis.3	⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))

Assertion

Ref	Expression
wfis	⊢ (𝑦 ∈ 𝐴 → 𝜑)

Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem wfis

Step	Hyp	Ref	Expression
1		wfis.1	. . 3 ⊢ 𝑅 We 𝐴
2		wfis.2	. . 3 ⊢ 𝑅 Se 𝐴
3		wfis.3	. . . 4 ⊢ (𝑦 ∈ 𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑 → 𝜑))
4	3	wfisg 6345	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
5	1, 2, 4	mp2an 689	. 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑
6	5	rspec 3239	1 ⊢ (𝑦 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2098  ∀wral 3053  [wsbc 3770   Se wse 5620   We wwe 5621  Predcpred 6290

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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

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-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-po 5579  df-so 5580  df-fr 5622  df-se 5623  df-we 5624  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291

This theorem is referenced by: (None)