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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 2100 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 692 | . 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴) |
3 | elsucg 4264 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 167 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 670 = wceq 1299 ∈ wcel 1448 suc csuc 4225 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-v 2643 df-un 3025 df-sn 3480 df-suc 4231 |
This theorem is referenced by: sucid 4277 nsuceq0g 4278 trsuc 4282 sucssel 4284 ordsucg 4356 sucunielr 4364 suc11g 4410 nlimsucg 4419 peano2b 4466 omsinds 4473 frecsuclem 6233 phplem4dom 6685 phplem4on 6690 dif1en 6702 fin0 6708 fin0or 6709 fidcenumlemrks 6769 bj-peano4 12738 |
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