Theorem alephsmo 8925

Description: The aleph function is strictly monotone. (Contributed by Mario Carneiro, 15-Mar-2013.)

Assertion

Ref	Expression
alephsmo	|- Smo aleph

Proof of Theorem alephsmo

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssid 3624	. 2 |- On C_ On
2		ordon 6982	. 2 |- Ord On
3		alephord2i 8900	. . . 4 |- ( x e. On -> ( y e. x -> ( aleph ` y ) e. ( aleph ` x ) ) )
4	3	ralrimiv 2965	. . 3 |- ( x e. On -> A. y e. x ( aleph ` y ) e. ( aleph ` x ) )
5	4	rgen 2922	. 2 |- A. x e. On A. y e. x ( aleph ` y ) e. ( aleph ` x )
6		alephfnon 8888	. . . 4 |- aleph Fn On
7		alephsson 8923	. . . 4 |- ran aleph C_ On
8		df-f 5892	. . . 4 |- ( aleph : On --> On <-> ( aleph Fn On /\ ran aleph C_ On ) )
9	6, 7, 8	mpbir2an 955	. . 3 |- aleph : On --> On
10		issmo2 7446	. . 3 |- ( aleph : On --> On -> ( ( On C_ On /\ Ord On /\ A. x e. On A. y e. x ( aleph ` y ) e. ( aleph ` x ) ) -> Smo aleph ) )
11	9, 10	ax-mp 5	. 2 |- ( ( On C_ On /\ Ord On /\ A. x e. On A. y e. x ( aleph ` y ) e. ( aleph ` x ) ) -> Smo aleph )
12	1, 2, 5, 11	mp3an 1424	1 |- Smo aleph

Syntax hints:   -> wi 4   /\ w3a 1037   e. wcel 1990   A.wral 2912   C_ wss 3574   ran crn 5115   Ord word 5722   Oncon0 5723   Fn wfn 5883   -->wf 5884   ` cfv 5888   Smo wsmo 7442   alephcale 8762

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766

This theorem is referenced by:  alephf1ALT  8926  alephsing  9098