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Mirrors > Home > MPE Home > Th. List > tfinds3 | Structured version Visualization version GIF version |
Description: Principle of Transfinite Induction (inference schema), using implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction step for successors, and the induction step for limit ordinals. (Contributed by NM, 6-Jan-2005.) (Revised by David Abernethy, 21-Jun-2011.) |
Ref | Expression |
---|---|
tfinds3.1 | ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) |
tfinds3.2 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
tfinds3.3 | ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) |
tfinds3.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
tfinds3.5 | ⊢ (𝜂 → 𝜓) |
tfinds3.6 | ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃))) |
tfinds3.7 | ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))) |
Ref | Expression |
---|---|
tfinds3 | ⊢ (𝐴 ∈ On → (𝜂 → 𝜏)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfinds3.1 | . . 3 ⊢ (𝑥 = ∅ → (𝜑 ↔ 𝜓)) | |
2 | 1 | imbi2d 343 | . 2 ⊢ (𝑥 = ∅ → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜓))) |
3 | tfinds3.2 | . . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) | |
4 | 3 | imbi2d 343 | . 2 ⊢ (𝑥 = 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜒))) |
5 | tfinds3.3 | . . 3 ⊢ (𝑥 = suc 𝑦 → (𝜑 ↔ 𝜃)) | |
6 | 5 | imbi2d 343 | . 2 ⊢ (𝑥 = suc 𝑦 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜃))) |
7 | tfinds3.4 | . . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) | |
8 | 7 | imbi2d 343 | . 2 ⊢ (𝑥 = 𝐴 → ((𝜂 → 𝜑) ↔ (𝜂 → 𝜏))) |
9 | tfinds3.5 | . 2 ⊢ (𝜂 → 𝜓) | |
10 | tfinds3.6 | . . 3 ⊢ (𝑦 ∈ On → (𝜂 → (𝜒 → 𝜃))) | |
11 | 10 | a2d 29 | . 2 ⊢ (𝑦 ∈ On → ((𝜂 → 𝜒) → (𝜂 → 𝜃))) |
12 | r19.21v 3177 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) ↔ (𝜂 → ∀𝑦 ∈ 𝑥 𝜒)) | |
13 | tfinds3.7 | . . . 4 ⊢ (Lim 𝑥 → (𝜂 → (∀𝑦 ∈ 𝑥 𝜒 → 𝜑))) | |
14 | 13 | a2d 29 | . . 3 ⊢ (Lim 𝑥 → ((𝜂 → ∀𝑦 ∈ 𝑥 𝜒) → (𝜂 → 𝜑))) |
15 | 12, 14 | syl5bi 244 | . 2 ⊢ (Lim 𝑥 → (∀𝑦 ∈ 𝑥 (𝜂 → 𝜒) → (𝜂 → 𝜑))) |
16 | 2, 4, 6, 8, 9, 11, 15 | tfinds 7576 | 1 ⊢ (𝐴 ∈ On → (𝜂 → 𝜏)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∅c0 4293 Oncon0 6193 Lim wlim 6194 suc csuc 6195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 |
This theorem is referenced by: oacl 8162 omcl 8163 oecl 8164 oawordri 8178 oaass 8189 oarec 8190 omordi 8194 omwordri 8200 odi 8207 omass 8208 oen0 8214 oewordri 8220 oeworde 8221 oeoelem 8226 omabs 8276 tfindsd 40571 |
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