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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 6183 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑦 ∈ 𝐴 𝜑) |
5 | 1, 2, 4 | mp2an 692 | . 2 ⊢ ∀𝑦 ∈ 𝐴 𝜑 |
6 | 5 | rspec 3119 | 1 ⊢ (𝑦 ∈ 𝐴 → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 ∀wral 3051 [wsbc 3683 Se wse 5492 We wwe 5493 Predcpred 6139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 |
This theorem is referenced by: (None) |
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