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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 4297 | . . . 4 | |
2 | sucidg 4171 | . . . . . . . . . . . 12 | |
3 | eleq2 2142 | . . . . . . . . . . . 12 | |
4 | 2, 3 | syl5ibrcom 155 | . . . . . . . . . . 11 |
5 | elsucg 4159 | . . . . . . . . . . 11 | |
6 | 4, 5 | sylibd 147 | . . . . . . . . . 10 |
7 | 6 | imp 122 | . . . . . . . . 9 |
8 | 7 | 3adant1 956 | . . . . . . . 8 |
9 | sucidg 4171 | . . . . . . . . . . . 12 | |
10 | eleq2 2142 | . . . . . . . . . . . 12 | |
11 | 9, 10 | syl5ibcom 153 | . . . . . . . . . . 11 |
12 | elsucg 4159 | . . . . . . . . . . 11 | |
13 | 11, 12 | sylibd 147 | . . . . . . . . . 10 |
14 | 13 | imp 122 | . . . . . . . . 9 |
15 | 14 | 3adant2 957 | . . . . . . . 8 |
16 | 8, 15 | jca 300 | . . . . . . 7 |
17 | eqcom 2083 | . . . . . . . . 9 | |
18 | 17 | orbi2i 711 | . . . . . . . 8 |
19 | 18 | anbi1i 445 | . . . . . . 7 |
20 | 16, 19 | sylib 120 | . . . . . 6 |
21 | ordir 763 | . . . . . 6 | |
22 | 20, 21 | sylibr 132 | . . . . 5 |
23 | 22 | ord 675 | . . . 4 |
24 | 1, 23 | mpi 15 | . . 3 |
25 | 24 | 3expia 1140 | . 2 |
26 | suceq 4157 | . 2 | |
27 | 25, 26 | impbid1 140 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-v 2603 df-dif 2975 df-un 2977 df-sn 3404 df-pr 3405 df-suc 4126 |
This theorem is referenced by: suc11 4301 peano4 4338 frecsuclem3 6013 |
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