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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 8550 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1 | ltp1i 8420 | . 2 ⊢ 3 < (3 + 1) |
3 | df-4 8537 | . 2 ⊢ 4 = (3 + 1) | |
4 | 2, 3 | breqtrri 3876 | 1 ⊢ 3 < 4 |
Colors of variables: wff set class |
Syntax hints: class class class wbr 3851 (class class class)co 5666 1c1 7405 + caddc 7407 < clt 7576 3c3 8528 4c4 8529 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-i2m1 7504 ax-0lt1 7505 ax-0id 7507 ax-rnegex 7508 ax-pre-ltadd 7515 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-xp 4457 df-iota 4993 df-fv 5036 df-ov 5669 df-pnf 7578 df-mnf 7579 df-ltxr 7581 df-2 8535 df-3 8536 df-4 8537 |
This theorem is referenced by: 2lt4 8643 3lt5 8646 3lt6 8651 3lt7 8657 3lt8 8664 3lt9 8672 3halfnz 8897 3lt10 9067 fldiv4p1lem1div2 9766 ef01bndlem 11101 sin01bnd 11102 flodddiv4 11266 srngstrd 11670 ex-fl 11918 |
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