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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-nnen2lp | Unicode version |
Description: A version of en2lp 4552 for natural numbers, which does not require
ax-setind 4535.
Note: using this theorem and bj-nnelirr 14556, one can remove dependency on ax-setind 4535 from nntri2 6491 and nndcel 6497; one can actually remove more dependencies from these. (Contributed by BJ, 28-Nov-2019.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-nnen2lp |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-nnelirr 14556 |
. . 3
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2 | 1 | adantl 277 |
. 2
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3 | bj-nntrans 14554 |
. . . . 5
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4 | 3 | adantl 277 |
. . . 4
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5 | ssel 3149 |
. . . 4
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6 | 4, 5 | syl6 33 |
. . 3
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7 | 6 | impd 254 |
. 2
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8 | 2, 7 | mtod 663 |
1
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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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-nul 4128 ax-pr 4208 ax-un 4432 ax-bd0 14416 ax-bdor 14419 ax-bdn 14420 ax-bdal 14421 ax-bdex 14422 ax-bdeq 14423 ax-bdel 14424 ax-bdsb 14425 ax-bdsep 14487 ax-infvn 14544 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-sn 3598 df-pr 3599 df-uni 3810 df-int 3845 df-suc 4370 df-iom 4589 df-bdc 14444 df-bj-ind 14530 |
This theorem is referenced by: bj-peano4 14558 |
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