Theorem alephord 3680

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 3679	. . 3 |- (B e. On -> (A e. B -> (aleph` A) ~< (aleph` B)))
2	1	adantl 305	. 2 |- ((A e. On /\ B e. On) -> (A e. B -> (aleph` A) ~< (aleph` B)))
3		alephordi 3679	. . . . . . . . 9 |- (A e. On -> (B e. A -> (aleph` B) ~< (aleph` A)))
4	3	con3d 87	. . . . . . . 8 |- (A e. On -> (-. (aleph` B) ~< (aleph` A) -> -. B e. A))
5		alephon 3671	. . . . . . . . 9 |- (aleph` A) e. On
6		alephon 3671	. . . . . . . . 9 |- (aleph` B) e. On
7		domtri 3644	. . . . . . . . 9 |- (((aleph` A) e. On /\ (aleph` B) e. On) -> ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A)))
8	5, 6, 7	mp2an 520	. . . . . . . 8 |- ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A))
9	4, 8	syl5ib 181	. . . . . . 7 |- (A e. On -> ((aleph` A) ~<_ (aleph` B) -> -. B e. A))
10	9	adantr 306	. . . . . 6 |- ((A e. On /\ B e. On) -> ((aleph` A) ~<_ (aleph` B) -> -. B e. A))
11		ontri1 2232	. . . . . 6 |- ((A e. On /\ B e. On) -> (A (_ B <-> -. B e. A))
12	10, 11	sylibrd 179	. . . . 5 |- ((A e. On /\ B e. On) -> ((aleph` A) ~<_ (aleph` B) -> A (_ B))
13		fveq2 2832	. . . . . . . 8 |- (A = B -> (aleph` A) = (aleph` B))
14		eqeng 3296	. . . . . . . . 9 |- ((aleph` A) e. On -> ((aleph` A) = (aleph` B) -> (aleph` A) ~~ (aleph` B)))
15	5, 14	ax-mp 6	. . . . . . . 8 |- ((aleph` A) = (aleph` B) -> (aleph` A) ~~ (aleph` B))
16	13, 15	syl 12	. . . . . . 7 |- (A = B -> (aleph` A) ~~ (aleph` B))
17	16	con3i 90	. . . . . 6 |- (-. (aleph` A) ~~ (aleph` B) -> -. A = B)
18	17	a1i 7	. . . . 5 |- ((A e. On /\ B e. On) -> (-. (aleph` A) ~~ (aleph` B) -> -. A = B))
19	12, 18	anim12d 431	. . . 4 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> (A (_ B /\ -. A = B)))
20		onelpsst 2253	. . . 4 |- ((A e. On /\ B e. On) -> (A e. B <-> (A (_ B /\ -. A = B)))
21	19, 20	sylibrd 179	. . 3 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> A e. B))
22		brsdom 3286	. . 3 |- ((aleph` A) ~< (aleph` B) <-> ((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)))
23	21, 22	syl5ib 181	. 2 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< (aleph` B) -> A e. B))
24	2, 23	impbid 397	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487   class class class wbr 2054  Oncon0 2199  ` cfv 2422   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273  alephcale 3621

This theorem is referenced by:  alephord2 3681

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-rdg 2970  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624

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