Theorem alephord2 8448

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 8447	. 2 |- ( ( A e. On /\ B e. On ) -> ( A e. B <-> ( aleph ` A ) ~< ( aleph ` B ) ) )
2		alephon 8441	. . . 4 |- ( aleph ` A ) e. On
3		alephon 8441	. . . . 5 |- ( aleph ` B ) e. On
4		onenon 8321	. . . . 5 |- ( ( aleph ` B ) e. On -> ( aleph ` B ) e. dom card )
5	3, 4	ax-mp 5	. . . 4 |- ( aleph ` B ) e. dom card
6		cardsdomel 8346	. . . 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 670	. . 3 |- ( ( aleph ` A ) ~< ( aleph ` B ) <-> ( aleph ` A ) e. ( card ` ( aleph ` B ) ) )
8		alephcard 8442	. . . 4 |- ( card ` ( aleph ` B ) ) = ( aleph ` B )
9	8	eleq2i 2532	. . 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 367   e. wcel 1823   class class class wbr 4439   Oncon0 4867   dom cdm 4988   ` cfv 5570   ~< csdm 7508   cardccrd 8307   alephcale 8308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-har 7976  df-card 8311  df-aleph 8312

This theorem is referenced by:  alephord2i  8449  alephord3  8450  alephiso  8470  alephval3  8482