Theorem wfis 6316

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 6315	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑)
5	1, 2, 4	mp2an 693	. 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑
6	5	rspec 3228	1 ⊢ (𝑦 ∈ 𝐴 → 𝜑)

Syntax hints:   → wi 4   ∈ wcel 2114  ∀wral 3051  [wsbc 3728   Se wse 5582   We wwe 5583  Predcpred 6264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265

This theorem is referenced by: (None)