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Mirrors > Home > MPE Home > Th. List > 3lt4 | Structured version Visualization version 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 11705 | . . 3 ⊢ 3 ∈ ℝ | |
2 | 1 | ltp1i 11533 | . 2 ⊢ 3 < (3 + 1) |
3 | df-4 11690 | . 2 ⊢ 4 = (3 + 1) | |
4 | 2, 3 | breqtrri 5057 | 1 ⊢ 3 < 4 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5030 (class class class)co 7135 1c1 10527 + caddc 10529 < clt 10664 3c3 11681 4c4 11682 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-2 11688 df-3 11689 df-4 11690 |
This theorem is referenced by: 2lt4 11800 3lt5 11803 3lt6 11808 3lt7 11814 3lt8 11821 3lt9 11829 3halfnz 12049 3lt10 12223 fz0to4untppr 13005 fldiv4p1lem1div2 13200 bpoly4 15405 ef01bndlem 15529 sin01bnd 15530 flodddiv4 15754 srngstr 16619 cnfldfun 20103 dveflem 24582 tangtx 25098 ppiublem1 25786 bpos1 25867 bposlem2 25869 gausslemma2dlem4 25953 2lgslem3b 25981 2lgslem3d 25983 chebbnd1lem2 26054 chebbnd1lem3 26055 chebbnd1 26056 pntlemb 26181 usgrexmplef 27049 upgr4cycl4dv4e 27970 ex-fl 28232 hlhilsmul 39237 stoweidlem26 42668 stoweid 42705 mod42tp1mod8 44120 nnsum4primes4 44307 nnsum4primesprm 44309 nnsum4primesgbe 44311 nnsum4primesle9 44313 nnsum4primeseven 44318 nnsum4primesevenALTV 44319 wtgoldbnnsum4prm 44320 ackval42 45110 |
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