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| Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nnen2lp | Unicode version | ||
| Description: A version of en2lp 4602 for natural numbers, which does not require
ax-setind 4585.
Note: using this theorem and bj-nnelirr 15889, one can remove dependency on ax-setind 4585 from nntri2 6580 and nndcel 6586; 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 15889 |
. . 3
| |
| 2 | 1 | adantl 277 |
. 2
|
| 3 | bj-nntrans 15887 |
. . . . 5
| |
| 4 | 3 | adantl 277 |
. . . 4
|
| 5 | ssel 3187 |
. . . 4
| |
| 6 | 4, 5 | syl6 33 |
. . 3
|
| 7 | 6 | impd 254 |
. 2
|
| 8 | 2, 7 | mtod 665 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-nul 4170 ax-pr 4253 ax-un 4480 ax-bd0 15749 ax-bdor 15752 ax-bdn 15753 ax-bdal 15754 ax-bdex 15755 ax-bdeq 15756 ax-bdel 15757 ax-bdsb 15758 ax-bdsep 15820 ax-infvn 15877 |
| This theorem depends on definitions: df-bi 117 df-tru 1376 df-nf 1484 df-sb 1786 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-sn 3639 df-pr 3640 df-uni 3851 df-int 3886 df-suc 4418 df-iom 4639 df-bdc 15777 df-bj-ind 15863 |
| This theorem is referenced by: bj-peano4 15891 |
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