Theorem rnss | index | src |

theorem rnss (A B: set): $ A C_ B -> Ran A C_ Ran B $;
StepHypRefExpression
1 equn1
Ran A C_ Ran B <-> Ran A u. Ran B == Ran B
2 equn1
A C_ B <-> A u. B == B
3 rnun
Ran (A u. B) == Ran A u. Ran B
4 rneq
A u. B == B -> Ran (A u. B) == Ran B
5 3, 4 syl5eqsr
A u. B == B -> Ran A u. Ran B == Ran B
6 2, 5 sylbi
A C_ B -> Ran A u. Ran B == Ran B
7 1, 6 sylibr
A C_ B -> Ran A C_ Ran 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_set (elab, ax_8), axs_the (theid, the0), axs_peano (peano2, addeq, muleq)