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Mirrors > Home > ILE Home > Th. List > subap0d | Unicode version |
Description: Two numbers apart from each other have difference apart from zero. (Contributed by Jim Kingdon, 12-Aug-2021.) (Proof shortened by BJ, 15-Aug-2024.) |
Ref | Expression |
---|---|
subap0d.a |
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subap0d.b |
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subap0d.ap |
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Ref | Expression |
---|---|
subap0d |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subap0d.ap |
. 2
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2 | subap0d.a |
. . 3
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3 | subap0d.b |
. . 3
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4 | subap0 8635 |
. . 3
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5 | 2, 3, 4 | syl2anc 411 |
. 2
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6 | 1, 5 | mpbird 167 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-id 4314 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-iota 5199 df-fun 5240 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-pnf 8029 df-mnf 8030 df-ltxr 8032 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 |
This theorem is referenced by: abssubap0 11140 climuni 11342 pwm1geoserap1 11557 geolim 11560 geolim2 11561 georeclim 11562 geoisum1c 11569 tanaddap 11788 cnopnap 14579 limcimo 14619 dvlemap 14634 dvconst 14646 dvid 14647 dvcnp2cntop 14648 dvaddxxbr 14650 dvmulxxbr 14651 dvcoapbr 14656 dvcjbr 14657 dvrecap 14662 dvef 14673 |
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