Theorem alephord2 8344

Description: Ordering property of the aleph function. Theorem 8A(a) of [Enderton] p. 213 and its converse. (Contributed by NM, 3-Nov-2003.) (Revised by Mario Carneiro, 9-Feb-2013.)

Assertion

Ref	Expression
alephord2	|- ( ( A e. On /\ B e. On ) -> ( A e. B <-> ( aleph ` A ) e. ( aleph ` B ) ) )

Proof of Theorem alephord2

Step	Hyp	Ref	Expression
1		alephord 8343	. 2 |- ( ( A e. On /\ B e. On ) -> ( A e. B <-> ( aleph ` A ) ~< ( aleph ` B ) ) )
2		alephon 8337	. . . 4 |- ( aleph ` A ) e. On
3		alephon 8337	. . . . 5 |- ( aleph ` B ) e. On
4		onenon 8217	. . . . 5 |- ( ( aleph ` B ) e. On -> ( aleph ` B ) e. dom card )
5	3, 4	ax-mp 5	. . . 4 |- ( aleph ` B ) e. dom card
6		cardsdomel 8242	. . . 4 |- ( ( ( aleph ` A ) e. On /\ ( aleph ` B ) e. dom card ) -> ( ( aleph ` A ) ~< ( aleph ` B ) <-> ( aleph ` A ) e. ( card ` ( aleph ` B ) ) ) )
7	2, 5, 6	mp2an 672	. . 3 |- ( ( aleph ` A ) ~< ( aleph ` B ) <-> ( aleph ` A ) e. ( card ` ( aleph ` B ) ) )
8		alephcard 8338	. . . 4 |- ( card ` ( aleph ` B ) ) = ( aleph ` B )
9	8	eleq2i 2527	. . 3 |- ( ( aleph ` A ) e. ( card ` ( aleph ` B ) ) <-> ( aleph ` A ) e. ( aleph ` B ) )
10	7, 9	bitri 249	. 2 |- ( ( aleph ` A ) ~< ( aleph ` B ) <-> ( aleph ` A ) e. ( aleph ` B ) )
11	1, 10	syl6bb 261	1 |- ( ( A e. On /\ B e. On ) -> ( A e. B <-> ( aleph ` A ) e. ( aleph ` B ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1758   class class class wbr 4387   Oncon0 4814   dom cdm 4935   ` cfv 5513   ~< csdm 7406   cardccrd 8203   alephcale 8204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-inf2 7945

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-se 4775  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-isom 5522  df-riota 6148  df-om 6574  df-recs 6929  df-rdg 6963  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-fin 7411  df-oi 7822  df-har 7871  df-card 8207  df-aleph 8208

This theorem is referenced by:  alephord2i  8345  alephord3  8346  alephiso  8366  alephval3  8378