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Mirrors > Home > ILE Home > Th. List > 6lt10 | GIF version |
Description: 6 is less than 10. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 8-Sep-2021.) |
Ref | Expression |
---|---|
6lt10 | ⊢ 6 < ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6lt7 9134 | . 2 ⊢ 6 < 7 | |
2 | 7lt10 9547 | . 2 ⊢ 7 < ;10 | |
3 | 6re 9031 | . . 3 ⊢ 6 ∈ ℝ | |
4 | 7re 9033 | . . 3 ⊢ 7 ∈ ℝ | |
5 | 10re 9433 | . . 3 ⊢ ;10 ∈ ℝ | |
6 | 3, 4, 5 | lttri 8093 | . 2 ⊢ ((6 < 7 ∧ 7 < ;10) → 6 < ;10) |
7 | 1, 2, 6 | mp2an 426 | 1 ⊢ 6 < ;10 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4018 0cc0 7842 1c1 7843 < clt 8023 6c6 9005 7c7 9006 ;cdc 9415 |
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 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 ax-pre-lttrn 7956 ax-pre-ltadd 7958 |
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-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-xp 4650 df-iota 5196 df-fv 5243 df-ov 5900 df-pnf 8025 df-mnf 8026 df-ltxr 8028 df-inn 8951 df-2 9009 df-3 9010 df-4 9011 df-5 9012 df-6 9013 df-7 9014 df-8 9015 df-9 9016 df-dec 9416 |
This theorem is referenced by: 5lt10 9549 plendxnvscandx 12723 slotsdnscsi 12733 |
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