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Theorem wfisg 6358
Description: Well-Ordered Induction Schema. If a property passes from all elements less than 𝑦 of a well-ordered class 𝐴 to 𝑦 itself (induction hypothesis), then the property holds for all elements of 𝐴. (Contributed by Scott Fenton, 11-Feb-2011.) (Proof shortened by Scott Fenton, 17-Nov-2024.)
Hypothesis
Ref Expression
wfisg.1 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
Assertion
Ref Expression
wfisg ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Distinct variable groups:   𝑦,𝐴,𝑧   𝜑,𝑧   𝑦,𝑅,𝑧
Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem wfisg
StepHypRef Expression
1 wefr 5664 . . 3 (𝑅 We 𝐴𝑅 Fr 𝐴)
21adantr 479 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Fr 𝐴)
3 weso 5665 . . . 4 (𝑅 We 𝐴𝑅 Or 𝐴)
4 sopo 5605 . . . 4 (𝑅 Or 𝐴𝑅 Po 𝐴)
53, 4syl 17 . . 3 (𝑅 We 𝐴𝑅 Po 𝐴)
65adantr 479 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Po 𝐴)
7 simpr 483 . 2 ((𝑅 We 𝐴𝑅 Se 𝐴) → 𝑅 Se 𝐴)
8 wfisg.1 . . . 4 (𝑦𝐴 → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
98adantl 480 . . 3 (((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) ∧ 𝑦𝐴) → (∀𝑧 ∈ Pred (𝑅, 𝐴, 𝑦)[𝑧 / 𝑦]𝜑𝜑))
109frpoinsg 6348 . 2 ((𝑅 Fr 𝐴𝑅 Po 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
112, 6, 7, 10syl3anc 1368 1 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑦𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084  wcel 2099  wral 3051  [wsbc 3775   Po wpo 5584   Or wor 5585   Fr wfr 5626   Se wse 5627   We wwe 5628  Predcpred 6303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-br 5146  df-opab 5208  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-cnv 5682  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304
This theorem is referenced by:  wfis  6360  wfis2fgOLD  6362
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