Theorem ltne | index | src |

theorem ltne (a b: nat): $ a < b -> a != b $;
StepHypRefExpression
1 ltirr
~b < b
2 lteq1
a = b -> (a < b <-> b < b)
3 2 bi1d
a = b -> a < b -> b < b
4 3 com12
a < b -> a = b -> b < b
5 4 con3d
a < b -> ~b < b -> ~a = b
6 5 conv ne
a < b -> ~b < b -> a != b
7 1, 6 mpi
a < b -> a != b

Axiom use

axs_prop_calc (ax_1, ax_2, ax_3, ax_mp, itru), axs_pred_calc (ax_gen, ax_4, ax_5, ax_6, ax_7, ax_10, ax_11, ax_12), axs_peano (peano1, peano2, peano5, addeq, add0, addS)