Theorem alephord 4847

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 4846	. . 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 4846	. . . . . . . . 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 4837	. . . . . . . . 9 |- (aleph` A) e. On
6		alephon 4837	. . . . . . . . 9 |- (aleph` B) e. On
7		domtri 4810	. . . . . . . . 9 |- (((aleph` A) e. On /\ (aleph` B) e. On) -> ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A)))
8	5, 6, 7	mp2an 695	. . . . . . . 8 |- ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A))
9	4, 8	syl5ib 206	. . . . . . 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 2971	. . . . . 6 |- ((A e. On /\ B e. On) -> (A (_ B <-> -. B e. A))
12	10, 11	sylibrd 204	. . . . 5 |- ((A e. On /\ B e. On) -> ((aleph` A) ~<_ (aleph` B) -> A (_ B))
13		fveq2 3709	. . . . . . . 8 |- (A = B -> (aleph` A) = (aleph` B))
14		eqeng 4373	. . . . . . . . 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 1599	. . . . . 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 556	. . . 4 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> (A (_ B /\ A =/= B)))
20		onelpsst 2988	. . . 4 |- ((A e. On /\ B e. On) -> (A e. B <-> (A (_ B /\ A =/= B)))
21	19, 20	sylibrd 204	. . 3 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> A e. B))
22		brsdom 4363	. . 3 |- ((aleph` A) ~< (aleph` B) <-> ((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)))
23	21, 22	syl5ib 206	. 2 |- ((A e. On /\ B e. On) -> ((aleph` A) ~< (aleph` B) -> A e. B))
24	2, 23	impbid 514	1 |- ((A e. On /\ B e. On) -> (A e. B <-> (aleph` A) ~< (aleph` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955   =/= wne 1577   (_ wss 2037   class class class wbr 2609  Oncon0 2938  ` cfv 3172   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350  alephcale 4786

This theorem is referenced by:  alephord2 4848  alephval2 4874

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353  df-card 4788  df-aleph 4789

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