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Mirrors > Home > MPE Home > Th. List > wfis | Structured version Visualization version GIF version |
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.) |
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)) |
Ref | Expression |
---|---|
wfis | ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
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 6354 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) |
5 | 1, 2, 4 | mp2an 691 | . 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑 |
6 | 5 | rspec 3243 | 1 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ∀wral 3057 [wsbc 3775 Se wse 5626 We wwe 5627 Predcpred 6299 |
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 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-br 5144 df-opab 5206 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-cnv 5681 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 |
This theorem is referenced by: (None) |
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