Theorem alephsucpw 4850

Description: The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous.)

Assertion

Ref	Expression
alephsucpw	|- (aleph` suc A) ~<_ P~(aleph` A)

Proof of Theorem alephsucpw

Step	Hyp	Ref	Expression
1		fvex 3723	. . . 4 |- (aleph` A) e. V
2	1	canth2 4470	. . 3 |- (aleph` A) ~< P~(aleph` A)
3		alephnbtwn2 4849	. . . 4 |- -. ((aleph` A) ~< P~(aleph` A) /\ P~(aleph` A) ~< (aleph` suc A))
4		imnan 242	. . . 4 |- (((aleph` A) ~< P~(aleph` A) -> -. P~(aleph` A) ~< (aleph` suc A)) <-> -. ((aleph` A) ~< P~(aleph` A) /\ P~(aleph` A) ~< (aleph` suc A)))
5	3, 4	mpbir 190	. . 3 |- ((aleph` A) ~< P~(aleph` A) -> -. P~(aleph` A) ~< (aleph` suc A))
6	2, 5	ax-mp 7	. 2 |- -. P~(aleph` A) ~< (aleph` suc A)
7		fvex 3723	. . 3 |- (aleph` suc A) e. V
8	1	pwex 2740	. . 3 |- P~(aleph` A) e. V
9		domtri 4818	. . 3 |- (((aleph` suc A) e. V /\ P~(aleph` A) e. V) -> ((aleph` suc A) ~<_ P~(aleph` A) <-> -. P~(aleph` A) ~< (aleph` suc A)))
10	7, 8, 9	mp2an 696	. 2 |- ((aleph` suc A) ~<_ P~(aleph` A) <-> -. P~(aleph` A) ~< (aleph` suc A))
11	6, 10	mpbir 190	1 |- (aleph` suc A) ~<_ P~(aleph` A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 956  Vcvv 1807  P~cpw 2397   class class class wbr 2614  suc csuc 2945  ` cfv 3177   ~<_ cdom 4355   ~< csdm 4356  alephcale 4794

This theorem is referenced by:  aleph1 4851

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-reg 4573  ax-inf2 4605  ax-ac 4724

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359  df-card 4796  df-aleph 4797

Copyright terms: Public domain