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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 2189 | . . 3 ⊢ 𝐴 = 𝐴 | |
2 | 1 | olci 733 | . 2 ⊢ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴) |
3 | elsucg 4425 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴 ∈ 𝐴 ∨ 𝐴 = 𝐴))) | |
4 | 2, 3 | mpbiri 168 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 709 = wceq 1364 ∈ wcel 2160 suc csuc 4386 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-sn 3616 df-suc 4392 |
This theorem is referenced by: sucid 4438 nsuceq0g 4439 trsuc 4443 sucssel 4445 ordsucg 4522 sucunielr 4530 suc11g 4577 nlimsucg 4586 peano2b 4635 omsinds 4642 nnpredlt 4644 frecsuclem 6435 phplem4dom 6894 phplem4on 6899 dif1en 6911 fin0 6917 fin0or 6918 fidcenumlemrks 6986 bj-peano4 15193 |
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