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Mirrors > Home > ILE Home > Th. List > uzind4ALT | GIF version |
Description: Induction on the upper set of integers that starts at an integer 𝑀. The last four hypotheses give us the substitution instances we need; the first two are the basis and the induction step. Either uzind4 9601 or uzind4ALT 9602 may be used; see comment for nnind 8948. (Contributed by NM, 7-Sep-2005.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
uzind4ALT.5 | ⊢ (𝑀 ∈ ℤ → 𝜓) |
uzind4ALT.6 | ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) |
uzind4ALT.1 | ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) |
uzind4ALT.2 | ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) |
uzind4ALT.3 | ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) |
uzind4ALT.4 | ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) |
Ref | Expression |
---|---|
uzind4ALT | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzind4ALT.1 | . 2 ⊢ (𝑗 = 𝑀 → (𝜑 ↔ 𝜓)) | |
2 | uzind4ALT.2 | . 2 ⊢ (𝑗 = 𝑘 → (𝜑 ↔ 𝜒)) | |
3 | uzind4ALT.3 | . 2 ⊢ (𝑗 = (𝑘 + 1) → (𝜑 ↔ 𝜃)) | |
4 | uzind4ALT.4 | . 2 ⊢ (𝑗 = 𝑁 → (𝜑 ↔ 𝜏)) | |
5 | uzind4ALT.5 | . 2 ⊢ (𝑀 ∈ ℤ → 𝜓) | |
6 | uzind4ALT.6 | . 2 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (𝜒 → 𝜃)) | |
7 | 1, 2, 3, 4, 5, 6 | uzind4 9601 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝜏) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1363 ∈ wcel 2158 ‘cfv 5228 (class class class)co 5888 1c1 7825 + caddc 7827 ℤcz 9266 ℤ≥cuz 9541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-distr 7928 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-cnre 7935 ax-pre-ltirr 7936 ax-pre-ltwlin 7937 ax-pre-lttrn 7938 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3or 980 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-int 3857 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fv 5236 df-riota 5844 df-ov 5891 df-oprab 5892 df-mpo 5893 df-pnf 8007 df-mnf 8008 df-xr 8009 df-ltxr 8010 df-le 8011 df-sub 8143 df-neg 8144 df-inn 8933 df-n0 9190 df-z 9267 df-uz 9542 |
This theorem is referenced by: (None) |
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