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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 2037 | . . 3 ⊢ A = A | |
2 | 1 | olci 650 | . 2 ⊢ (A ∈ A ∨ A = A) |
3 | elsucg 4107 | . 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 628 = wceq 1242 ∈ wcel 1390 suc csuc 4068 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-sn 3373 df-suc 4074 |
This theorem is referenced by: sucid 4120 nsuceq0g 4121 trsuc 4125 sucssel 4127 ordsucg 4194 sucunielr 4201 suc11g 4235 nlimsucg 4242 onpsssuc 4247 peano2b 4280 frecsuclem2 5928 bj-peano4 9415 |
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