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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 7349 | . . 3 ⊢ ω ⊆ On | |
2 | epweon 7262 | . . 3 ⊢ E We On | |
3 | wess 5344 | . . 3 ⊢ (ω ⊆ On → ( E We On → E We ω)) | |
4 | 1, 2, 3 | mp2 9 | . 2 ⊢ E We ω |
5 | epse 5340 | . 2 ⊢ E Se ω | |
6 | omsinds.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | omsinds.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
8 | predep 5961 | . . . . 5 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = (ω ∩ 𝑥)) | |
9 | ordom 7354 | . . . . . . 7 ⊢ Ord ω | |
10 | ordtr 5992 | . . . . . . 7 ⊢ (Ord ω → Tr ω) | |
11 | trss 4998 | . . . . . . 7 ⊢ (Tr ω → (𝑥 ∈ ω → 𝑥 ⊆ ω)) | |
12 | 9, 10, 11 | mp2b 10 | . . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ ω) |
13 | sseqin2 4040 | . . . . . 6 ⊢ (𝑥 ⊆ ω ↔ (ω ∩ 𝑥) = 𝑥) | |
14 | 12, 13 | sylib 210 | . . . . 5 ⊢ (𝑥 ∈ ω → (ω ∩ 𝑥) = 𝑥) |
15 | 8, 14 | eqtrd 2814 | . . . 4 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥) |
16 | 15 | raleqdv 3340 | . . 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 5976 | 1 ⊢ (𝐴 ∈ ω → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ∩ cin 3791 ⊆ wss 3792 Tr wtr 4989 E cep 5267 We wwe 5315 Predcpred 5934 Ord word 5977 Oncon0 5978 ωcom 7345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-tr 4990 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-om 7346 |
This theorem is referenced by: (None) |
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