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 7069 | . . 3 ⊢ ω ⊆ On | |
2 | epweon 6983 | . . 3 ⊢ E We On | |
3 | wess 5101 | . . 3 ⊢ (ω ⊆ On → ( E We On → E We ω)) | |
4 | 1, 2, 3 | mp2 9 | . 2 ⊢ E We ω |
5 | epse 5097 | . 2 ⊢ E Se ω | |
6 | omsinds.1 | . 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | omsinds.2 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜒)) | |
8 | predep 5706 | . . . . 5 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = (ω ∩ 𝑥)) | |
9 | ordom 7074 | . . . . . . 7 ⊢ Ord ω | |
10 | ordtr 5737 | . . . . . . 7 ⊢ (Ord ω → Tr ω) | |
11 | trss 4761 | . . . . . . 7 ⊢ (Tr ω → (𝑥 ∈ ω → 𝑥 ⊆ ω)) | |
12 | 9, 10, 11 | mp2b 10 | . . . . . 6 ⊢ (𝑥 ∈ ω → 𝑥 ⊆ ω) |
13 | sseqin2 3817 | . . . . . 6 ⊢ (𝑥 ⊆ ω ↔ (ω ∩ 𝑥) = 𝑥) | |
14 | 12, 13 | sylib 208 | . . . . 5 ⊢ (𝑥 ∈ ω → (ω ∩ 𝑥) = 𝑥) |
15 | 8, 14 | eqtrd 2656 | . . . 4 ⊢ (𝑥 ∈ ω → Pred( E , ω, 𝑥) = 𝑥) |
16 | 15 | raleqdv 3144 | . . 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 5721 | 1 ⊢ (𝐴 ∈ ω → 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∩ cin 3573 ⊆ wss 3574 Tr wtr 4752 E cep 5028 We wwe 5072 Predcpred 5679 Ord word 5722 Oncon0 5723 ωcom 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |