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Mirrors > Home > ILE Home > Th. List > suc11g | Unicode version |
Description: The successor operation behaves like a one-to-one function (assuming the Axiom of Set Induction). Similar to Exercise 35 of [Enderton] p. 208 and its converse. (Contributed by NM, 25-Oct-2003.) |
Ref | Expression |
---|---|
suc11g |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en2lp 4469 | . . . 4 | |
2 | sucidg 4338 | . . . . . . . . . . . 12 | |
3 | eleq2 2203 | . . . . . . . . . . . 12 | |
4 | 2, 3 | syl5ibrcom 156 | . . . . . . . . . . 11 |
5 | elsucg 4326 | . . . . . . . . . . 11 | |
6 | 4, 5 | sylibd 148 | . . . . . . . . . 10 |
7 | 6 | imp 123 | . . . . . . . . 9 |
8 | 7 | 3adant1 999 | . . . . . . . 8 |
9 | sucidg 4338 | . . . . . . . . . . . 12 | |
10 | eleq2 2203 | . . . . . . . . . . . 12 | |
11 | 9, 10 | syl5ibcom 154 | . . . . . . . . . . 11 |
12 | elsucg 4326 | . . . . . . . . . . 11 | |
13 | 11, 12 | sylibd 148 | . . . . . . . . . 10 |
14 | 13 | imp 123 | . . . . . . . . 9 |
15 | 14 | 3adant2 1000 | . . . . . . . 8 |
16 | 8, 15 | jca 304 | . . . . . . 7 |
17 | eqcom 2141 | . . . . . . . . 9 | |
18 | 17 | orbi2i 751 | . . . . . . . 8 |
19 | 18 | anbi1i 453 | . . . . . . 7 |
20 | 16, 19 | sylib 121 | . . . . . 6 |
21 | ordir 806 | . . . . . 6 | |
22 | 20, 21 | sylibr 133 | . . . . 5 |
23 | 22 | ord 713 | . . . 4 |
24 | 1, 23 | mpi 15 | . . 3 |
25 | 24 | 3expia 1183 | . 2 |
26 | suceq 4324 | . 2 | |
27 | 25, 26 | impbid1 141 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wo 697 w3a 962 wceq 1331 wcel 1480 csuc 4287 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-dif 3073 df-un 3075 df-sn 3533 df-pr 3534 df-suc 4293 |
This theorem is referenced by: suc11 4473 peano4 4511 frecsuclem 6303 |
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