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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.) |
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 7330 | . . 3 ⊢ ω ⊆ On | |
2 | epweon 7243 | . . 3 ⊢ E We On | |
3 | wess 5329 | . . 3 ⊢ (ω ⊆ On → ( E We On → E We ω)) | |
4 | 1, 2, 3 | mp2 9 | . 2 ⊢ E We ω |
5 | epse 5325 | . 2 ⊢ E Se ω | |
6 | omsinds.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | omsinds.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
8 | predep 5946 | . . . . 5 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = (ω ∩ 𝑥)) | |
9 | ordom 7335 | . . . . . . 7 ⊢ Ord ω | |
10 | ordtr 5977 | . . . . . . 7 ⊢ (Ord ω → Tr ω) | |
11 | trss 4984 | . . . . . . 7 ⊢ (Tr ω → (𝑥 ∈ ω → 𝑥 ⊆ ω)) | |
12 | 9, 10, 11 | mp2b 10 | . . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ ω) |
13 | sseqin2 4044 | . . . . . 6 ⊢ (𝑥 ⊆ ω ↔ (ω ∩ 𝑥) = 𝑥) | |
14 | 12, 13 | sylib 210 | . . . . 5 ⊢ (𝑥 ∈ ω → (ω ∩ 𝑥) = 𝑥) |
15 | 8, 14 | eqtrd 2861 | . . . 4 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥) |
16 | 15 | raleqdv 3356 | . . 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 232 | . 2 ⊢ (𝑥 ∈ ω → (∀𝑦 ∈ Pred ( E , ω, 𝑥)𝜓 → 𝜑)) |
19 | 4, 5, 6, 7, 18 | wfis3 5961 | 1 ⊢ (𝐴 ∈ ω → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ∩ cin 3797 ⊆ wss 3798 Tr wtr 4975 E cep 5254 We wwe 5300 Predcpred 5919 Ord word 5962 Oncon0 5963 ωcom 7326 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-tr 4976 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-om 7327 |
This theorem is referenced by: (None) |
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