Theorem alephord 5025

Description: Ordering property of the aleph function.

Assertion

Ref	Expression
alephord	|- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))

Proof of Theorem alephord

Step	Hyp	Ref	Expression
1		alephordi 5024	. . 3 |- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))
2	1	adantl 388	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) ~< (aleph` B)))
3		alephordi 5024	. . . . . . . . 9 |- (A e. On -> (B e. A -> (aleph` B) ~< (aleph` A)))
4	3	con3d 95	. . . . . . . 8 |- (A e. On -> (-. (aleph` B) ~< (aleph` A) -> -. B e. A))
5		alephon 5015	. . . . . . . . 9 |- (aleph` A) e. On
6		alephon 5015	. . . . . . . . 9 |- (aleph` B) e. On
7		domtri 4987	. . . . . . . . 9 |- (((aleph` A) e. On /\ (aleph` B) e. On) -> ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A)))
8	5, 6, 7	mp2an 701	. . . . . . . 8 |- ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A))
9	4, 8	syl5ib 204	. . . . . . 7 |- (A e. On -> ((aleph` A) ~<_ (aleph` B) -> -. B e. A))
10	9	adantr 389	. . . . . 6 |- ((A e. On /\ B e. On) -> ((aleph` A) ~<_ (aleph` B) -> -. B e. A))
11		ontri1 3009	. . . . . 6 |- ((A e. On /\ B e. On) -> (A (_ B <-> -. B e. A))
12	10, 11	sylibrd 202	. . . . 5 |- ((A e. On /\ B e. On) -> ((aleph` A) ~<_ (aleph` B) -> A (_ B))
13		fveq2 3835	. . . . . . . 8 |- (A = B -> (aleph` A) = (aleph` B))
14		eqeng 4533	. . . . . . . . 9 |- ((aleph` A) e. On -> ((aleph` A) = (aleph` B) -> (aleph` A) ~~ (aleph` B)))
15	5, 14	ax-mp 7	. . . . . . . 8 |- ((aleph` A) = (aleph` B) -> (aleph` A) ~~ (aleph` B))
16	13, 15	syl 10	. . . . . . 7 |- (A = B -> (aleph` A) ~~ (aleph` B))
17	16	necon3bi 1650	. . . . . 6 |- (-. (aleph` A) ~~ (aleph` B) -> A =/= B)
18	17	a1i 8	. . . . 5 |- ((A e. On /\ B e. On) -> (-. (aleph` A) ~~ (aleph` B) -> A =/= B))
19	12, 18	anim12d 561	. . . 4 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> (A (_ B /\ A =/= B)))
20		onelpss 3015	. . . 4 |- ((A e. On /\ B e. On) -> (A e. B <-> (A (_ B /\ A =/= B)))
21	19, 20	sylibrd 202	. . 3 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> A e. B))
22		brsdom 4522	. . 3 |- ((aleph` A) ~< (aleph` B) <-> ((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)))
23	21, 22	syl5ib 204	. 2 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< (aleph` B) -> A e. B))
24	2, 23	impbid 519	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221   = wceq 992   e. wcel 994   =/= wne 1628   (_ wss 2099   class class class wbr 2692  Oncon0 2975  ` cfv 3263   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507  alephcale 4960

This theorem is referenced by:  alephord2 5026  alephval2 5052

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-rdg 4233  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511  df-card 4962  df-aleph 4963

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