![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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.) |
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 7222 | . . 3 ⊢ ω ⊆ On | |
2 | epweon 7136 | . . 3 ⊢ E We On | |
3 | wess 5241 | . . 3 ⊢ (ω ⊆ On → ( E We On → E We ω)) | |
4 | 1, 2, 3 | mp2 9 | . 2 ⊢ E We ω |
5 | epse 5237 | . 2 ⊢ E Se ω | |
6 | omsinds.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | omsinds.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
8 | predep 5855 | . . . . 5 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = (ω ∩ 𝑥)) | |
9 | ordom 7227 | . . . . . . 7 ⊢ Ord ω | |
10 | ordtr 5886 | . . . . . . 7 ⊢ (Ord ω → Tr ω) | |
11 | trss 4901 | . . . . . . 7 ⊢ (Tr ω → (𝑥 ∈ ω → 𝑥 ⊆ ω)) | |
12 | 9, 10, 11 | mp2b 10 | . . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ ω) |
13 | sseqin2 3948 | . . . . . 6 ⊢ (𝑥 ⊆ ω ↔ (ω ∩ 𝑥) = 𝑥) | |
14 | 12, 13 | sylib 208 | . . . . 5 ⊢ (𝑥 ∈ ω → (ω ∩ 𝑥) = 𝑥) |
15 | 8, 14 | eqtrd 2782 | . . . 4 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥) |
16 | 15 | raleqdv 3271 | . . 3 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 ↔ ∀𝑦 ∈ 𝑥 𝜓)) |
17 | omsinds.3 | . . 3 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ 𝑥 𝜓 → 𝜑)) | |
18 | 16, 17 | sylbid 230 | . 2 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 → 𝜑)) |
19 | 4, 5, 6, 7, 18 | wfis3 5870 | 1 ⊢ (𝐴 ∈ ω → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1620 ∈ wcel 2127 ∀wral 3038 ∩ cin 3702 ⊆ wss 3703 Tr wtr 4892 E cep 5166 We wwe 5212 Predcpred 5828 Ord word 5871 Oncon0 5872 ωcom 7218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pr 5043 ax-un 7102 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-br 4793 df-opab 4853 df-tr 4893 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-se 5214 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-om 7219 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |