![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 3lt5 | Structured version Visualization version GIF version |
Description: 3 is less than 5. (Contributed by Mario Carneiro, 15-Sep-2013.) |
Ref | Expression |
---|---|
3lt5 | ⊢ 3 < 5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3lt4 12327 | . 2 ⊢ 3 < 4 | |
2 | 4lt5 12330 | . 2 ⊢ 4 < 5 | |
3 | 3re 12233 | . . 3 ⊢ 3 ∈ ℝ | |
4 | 4re 12237 | . . 3 ⊢ 4 ∈ ℝ | |
5 | 5re 12240 | . . 3 ⊢ 5 ∈ ℝ | |
6 | 3, 4, 5 | lttri 11281 | . 2 ⊢ ((3 < 4 ∧ 4 < 5) → 3 < 5) |
7 | 1, 2, 6 | mp2an 690 | 1 ⊢ 3 < 5 |
Colors of variables: wff setvar class |
Syntax hints: class class class wbr 5105 < clt 11189 3c3 12209 4c4 12210 5c5 12211 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-po 5545 df-so 5546 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-2 12216 df-3 12217 df-4 12218 df-5 12219 |
This theorem is referenced by: 23prm 16991 43prm 16994 83prm 16995 163prm 16997 scandxnmulrndx 17199 ipsstr 17217 sramulrOLD 20645 zlmmulrOLD 20924 psrvalstr 21318 matscaOLD 21763 bpos1 26631 bposlem3 26634 cyc3conja 32006 resvmulrOLD 32131 algstr 41490 mnringscadOLD 42493 31prm 45779 sbgoldbo 45969 |
Copyright terms: Public domain | W3C validator |