Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > peano1 | Unicode version |
Description: Zero is a natural number. One of Peano's five postulates for arithmetic. Proposition 7.30(1) of [TakeutiZaring] p. 42. (Contributed by NM, 15-May-1994.) |
Ref | Expression |
---|---|
peano1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4116 | . . . 4 | |
2 | 1 | elint 3837 | . . 3 |
3 | df-clab 2157 | . . . 4 | |
4 | simpl 108 | . . . . . 6 | |
5 | 4 | sbimi 1757 | . . . . 5 |
6 | clelsb2 2276 | . . . . 5 | |
7 | 5, 6 | sylib 121 | . . . 4 |
8 | 3, 7 | sylbi 120 | . . 3 |
9 | 2, 8 | mpgbir 1446 | . 2 |
10 | dfom3 4576 | . 2 | |
11 | 9, 10 | eleqtrri 2246 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wsb 1755 wcel 2141 cab 2156 wral 2448 c0 3414 cint 3831 csuc 4350 com 4574 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 ax-nul 4115 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-nul 3415 df-int 3832 df-iom 4575 |
This theorem is referenced by: peano5 4582 limom 4598 nnregexmid 4605 omsinds 4606 nnpredcl 4607 frec0g 6376 frecabcl 6378 frecrdg 6387 oa1suc 6446 nna0r 6457 nnm0r 6458 nnmcl 6460 nnmsucr 6467 1onn 6499 nnm1 6504 nnaordex 6507 nnawordex 6508 php5 6836 php5dom 6841 0fin 6862 findcard2 6867 findcard2s 6868 infm 6882 inffiexmid 6884 0ct 7084 ctmlemr 7085 ctssdclemn0 7087 ctssdc 7090 omct 7094 nninfisol 7109 fodjum 7122 fodju0 7123 ctssexmid 7126 nninfwlpoimlemg 7151 nninfwlpoimlemginf 7152 1lt2pi 7302 nq0m0r 7418 nq0a0 7419 prarloclem5 7462 frec2uzrand 10361 frecuzrdg0 10369 frecuzrdg0t 10378 frecfzennn 10382 0tonninf 10395 1tonninf 10396 hashinfom 10712 hashunlem 10739 hash1 10746 ennnfonelemj0 12356 ennnfonelem1 12362 ennnfonelemhf1o 12368 ennnfonelemhom 12370 bj-nn0suc 13999 bj-nn0sucALT 14013 012of 14028 2o01f 14029 pwle2 14031 pwf1oexmid 14032 subctctexmid 14034 peano3nninf 14040 nninfall 14042 nninfsellemdc 14043 nninfsellemeq 14047 nninffeq 14053 isomninnlem 14062 iswomninnlem 14081 ismkvnnlem 14084 |
Copyright terms: Public domain | W3C validator |