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Mirrors > Home > ILE Home > Th. List > 3lt4 | GIF version |
Description: 3 is less than 4. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
3lt4 | ⊢ 3 < 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3re 9006 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1 | ltp1i 8875 | . 2 ⊢ 3 < (3 + 1) |
3 | df-4 8993 | . 2 ⊢ 4 = (3 + 1) | |
4 | 2, 3 | breqtrri 4042 | 1 ⊢ 3 < 4 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 4015 (class class class)co 5888 1c1 7825 + caddc 7827 < clt 8005 3c3 8984 4c4 8985 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 ax-setind 4548 ax-cnex 7915 ax-resscn 7916 ax-1cn 7917 ax-1re 7918 ax-icn 7919 ax-addcl 7920 ax-addrcl 7921 ax-mulcl 7922 ax-addcom 7924 ax-addass 7926 ax-i2m1 7929 ax-0lt1 7930 ax-0id 7932 ax-rnegex 7933 ax-pre-ltadd 7940 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-nel 2453 df-ral 2470 df-rex 2471 df-rab 2474 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-xp 4644 df-iota 5190 df-fv 5236 df-ov 5891 df-pnf 8007 df-mnf 8008 df-ltxr 8010 df-2 8991 df-3 8992 df-4 8993 |
This theorem is referenced by: 2lt4 9105 3lt5 9108 3lt6 9113 3lt7 9119 3lt8 9126 3lt9 9134 3halfnz 9363 3lt10 9533 fz0to4untppr 10137 fldiv4p1lem1div2 10318 ef01bndlem 11777 sin01bnd 11778 flodddiv4 11952 starvndxnmulrndx 12616 srngstrd 12618 dveflem 14458 tangtx 14530 ex-fl 14748 |
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