Theorem alephord 4875

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 4874	. . 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 4874	. . . . . . . . 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 4865	. . . . . . . . 9 |- (aleph` A) e. On
6		alephon 4865	. . . . . . . . 9 |- (aleph` B) e. On
7		domtri 4838	. . . . . . . . 9 |- (((aleph` A) e. On /\ (aleph` B) e. On) -> ((aleph` A) ~<_ (aleph` B) <-> -. (aleph` B) ~< (aleph` A)))
8	5, 6, 7	mp2an 697	. . . . . . . 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 2981	. . . . . 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 3724	. . . . . . . 8 |- (A = B -> (aleph` A) = (aleph` B))
14		eqeng 4392	. . . . . . . . 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 1607	. . . . . 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 558	. . . 4 |- ((A e. On /\ B e. On) -> (((aleph` A) ~<_ (aleph` B) /\ -. (aleph` A) ~~ (aleph` B)) -> (A (_ B /\ A =/= B)))
20		onelpsst 2998	. . . 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 4381	. . 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 516	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 956   e. wcel 958   =/= wne 1585   (_ wss 2047   class class class wbr 2619  Oncon0 2948  ` cfv 3182   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366  alephcale 4814

This theorem is referenced by:  alephord2 4876  alephval2 4902

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816  df-aleph 4817

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