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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nnen2lp | Unicode version |
Description: A version of en2lp 4439 for natural numbers, which does not require
ax-setind 4422.
Note: using this theorem and bj-nnelirr 13078, one can remove dependency on ax-setind 4422 from nntri2 6358 and nndcel 6364; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnen2lp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnelirr 13078 | . . 3 | |
2 | 1 | adantl 275 | . 2 |
3 | bj-nntrans 13076 | . . . . 5 | |
4 | 3 | adantl 275 | . . . 4 |
5 | ssel 3061 | . . . 4 | |
6 | 4, 5 | syl6 33 | . . 3 |
7 | 6 | impd 252 | . 2 |
8 | 2, 7 | mtod 637 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wcel 1465 wss 3041 com 4474 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-nul 4024 ax-pr 4101 ax-un 4325 ax-bd0 12938 ax-bdor 12941 ax-bdn 12942 ax-bdal 12943 ax-bdex 12944 ax-bdeq 12945 ax-bdel 12946 ax-bdsb 12947 ax-bdsep 13009 ax-infvn 13066 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-sn 3503 df-pr 3504 df-uni 3707 df-int 3742 df-suc 4263 df-iom 4475 df-bdc 12966 df-bj-ind 13052 |
This theorem is referenced by: bj-peano4 13080 |
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