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Mirrors > Home > ILE Home > Th. List > omex | Unicode version |
Description: The existence of omega (the class of natural numbers). Axiom 7 of [TakeutiZaring] p. 43. (Contributed by NM, 6-Aug-1994.) |
Ref | Expression |
---|---|
omex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zfinf2 4560 | . . 3 | |
2 | intexabim 4125 | . . 3 | |
3 | 1, 2 | ax-mp 5 | . 2 |
4 | dfom3 4563 | . . 3 | |
5 | 4 | eleq1i 2230 | . 2 |
6 | 3, 5 | mpbir 145 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wex 1479 wcel 2135 cab 2150 wral 2442 cvv 2721 c0 3404 cint 3818 csuc 4337 com 4561 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-in 3117 df-ss 3124 df-int 3819 df-iom 4562 |
This theorem is referenced by: peano5 4569 omelon 4580 frecex 6353 frecabex 6357 fict 6825 infnfi 6852 ominf 6853 inffiexmid 6863 omp1eom 7051 difinfsn 7056 0ct 7063 ctmlemr 7064 ctssdclemn0 7066 ctssdclemr 7068 ctssdc 7069 enumct 7071 omct 7073 ctfoex 7074 nninfex 7077 infnninf 7079 infnninfOLD 7080 nnnninf 7081 exmidlpo 7098 cc2lem 7198 niex 7244 enq0ex 7371 nq0ex 7372 uzenom 10350 frecfzennn 10351 nnenom 10359 fxnn0nninf 10363 0tonninf 10364 1tonninf 10365 inftonninf 10366 hashinfuni 10679 hashinfom 10680 xpct 12272 ennnfonelemj0 12277 ennnfonelemg 12279 ennnfonelemen 12297 ctiunct 12316 omctfn 12319 ssomct 12321 bj-charfunbi 13534 subctctexmid 13722 0nninf 13725 nnsf 13726 peano4nninf 13727 peano3nninf 13728 nninfself 13734 nninfsellemeq 13735 nninfsellemeqinf 13737 sbthom 13746 |
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