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Mirrors > Home > ILE Home > Th. List > sucidg | GIF version |
Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.) |
Ref | Expression |
---|---|
sucidg | ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2040 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 651 | . 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴) |
3 | elsucg 4141 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 157 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 629 = wceq 1243 ∈ wcel 1393 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-v 2559 df-un 2922 df-sn 3381 df-suc 4108 |
This theorem is referenced by: sucid 4154 nsuceq0g 4155 trsuc 4159 sucssel 4161 ordsucg 4228 sucunielr 4236 suc11g 4281 nlimsucg 4290 onpsssuc 4295 peano2b 4337 frecsuclem2 5989 phplem4dom 6324 phplem4on 6329 dif1en 6337 fin0 6342 fin0or 6343 bj-peano4 10080 |
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