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Mirrors > Home > MPE Home > Th. List > omsinds | Structured version Visualization version GIF version |
Description: Strong (or "total") induction principle over the finite ordinals. (Contributed by Scott Fenton, 17-Jul-2015.) (Proof shortened by BJ, 16-Oct-2024.) |
Ref | Expression |
---|---|
omsinds.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
omsinds.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) |
omsinds.3 | ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) |
Ref | Expression |
---|---|
omsinds | ⊢ (𝐴 ∈ ω → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omsson 7872 | . . 3 ⊢ ω ⊆ On | |
2 | epweon 7775 | . . 3 ⊢ E We On | |
3 | wess 5661 | . . 3 ⊢ (ω ⊆ On → ( E We On → E We ω)) | |
4 | 1, 2, 3 | mp2 9 | . 2 ⊢ E We ω |
5 | epse 5657 | . 2 ⊢ E Se ω | |
6 | omsinds.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | omsinds.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
8 | trom 7877 | . . . . 5 ⊢ Tr ω | |
9 | trpred 6336 | . . . . 5 ⊢ ((Tr ω ∧ 𝑥 ∈ ω) → Pred( E , ω, 𝑥) = 𝑥) | |
10 | 8, 9 | mpan 688 | . . . 4 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥) |
11 | 10 | raleqdv 3315 | . . 3 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 ↔ ∀𝑦 ∈ 𝑥 𝜓)) |
12 | omsinds.3 | . . 3 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
13 | 11, 12 | sylbid 239 | . 2 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 → 𝜑)) |
14 | 4, 5, 6, 7, 13 | wfis3 6366 | 1 ⊢ (𝐴 ∈ ω → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ⊆ wss 3946 Tr wtr 5262 E cep 5577 We wwe 5628 Predcpred 6303 Oncon0 6368 ωcom 7868 |
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-3or 1085 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-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-tr 5263 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-se 5630 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-om 7869 |
This theorem is referenced by: (None) |
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