Theorem sizeeqd ≪ | index | src | ≫

theorem sizeeqd (_G: wff) (_A1 _A2: set):
  $ _G -> _A1 == _A2 $ >
  $ _G -> size _A1 = size _A2 $;


_G -> _A1 == _A2
_G -> upto k == upto k
_G -> (_A1 C_ upto k <-> _A2 C_ upto k)
_G -> {k | _A1 C_ upto k} == {k | _A2 C_ upto k}
_G -> least {k | _A1 C_ upto k} = least {k | _A2 C_ upto k}
_G -> size _A1 = size _A2

Step	Hyp	Ref	Expression
1		hyp _Ah	_G -> _A1 == _A2
2		eqsidd	_G -> upto k == upto k
3	1, 2	sseqd	_G -> (_A1 C_ upto k <-> _A2 C_ upto k)
4	3	abeqd	_G -> {k \| _A1 C_ upto k} == {k \| _A2 C_ upto k}
5	4	leasteqd	_G -> least {k \| _A1 C_ upto k} = least {k \| _A2 C_ upto k}
6	5	conv size	_G -> size _A1 = size _A2

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)