Theorem alephord 6023

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 6022	. . 3 |- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))
2	1	adantl 424	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) ~< (aleph` B)))
3		alephordi 6022	. . . . . . . . 9 |- (A e. On -> (B e. A -> (aleph` B) ~< (aleph` A)))
4	3	con3d 111	. . . . . . . 8 |- (A e. On -> (-. (aleph` B) ~< (aleph` A) -> -. B e. A))
5		alephon 5876	. . . . . . . . 9 |- (aleph` A) e. On
6		alephon 5876	. . . . . . . . 9 |- (aleph` B) e. On
7		domtri 5989	. . . . . . . . 9 |- (((aleph` A) e. On /\ (aleph` B) e. On) -> ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A)))
8	5, 6, 7	mp2an 761	. . . . . . . 8 |- ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A))
9	4, 8	syl5ib 223	. . . . . . 7 |- (A e. On -> ((aleph` A) ~<_ (aleph` B) -> -. B e. A))
10	9	adantr 425	. . . . . 6 |- ((A e. On /\ B e. On) -> ((aleph` A) ~<_ (aleph` B) -> -. B e. A))
11		ontri1 3695	. . . . . 6 |- ((A e. On /\ B e. On) -> (A C_ B <-> -. B e. A))
12	10, 11	sylibrd 221	. . . . 5 |- ((A e. On /\ B e. On) -> ((aleph` A) ~<_ (aleph` B) -> A C_ B))
13		fveq2 4681	. . . . . . . 8 |- (A = B -> (aleph` A) = (aleph` B))
14		eqeng 5451	. . . . . . . . 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 12	. . . . . . 7 |- (A = B -> (aleph` A) ~~ (aleph` B))
17	16	necon3bi 2045	. . . . . 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 617	. . . 4 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> (A C_ B /\ A =/= B)))
20		onelpss 3703	. . . 4 |- ((A e. On /\ B e. On) -> (A e. B <-> (A C_ B /\ A =/= B)))
21	19, 20	sylibrd 221	. . 3 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> A e. B))
22		brsdom 5440	. . 3 |- ((aleph` A) ~< (aleph` B) <-> ((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)))
23	21, 22	syl5ib 223	. 2 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< (aleph` B) -> A e. B))
24	2, 23	impbid 574	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017   C_ wss 2593   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425  alephcale 5860

This theorem is referenced by:  alephord2 6024  alephval2 6050

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-rdg 5140  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-card 5862  df-aleph 5863

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