Theorem alephon 4845

Description: An aleph is an ordinal number.

Assertion

Ref	Expression
alephon	|- (aleph` A) e. On

Proof of Theorem alephon

Step	Hyp	Ref	Expression
1		fveq2 3715	. . . 4 |- (x = (/) -> (aleph` x) = (aleph` (/)))
2	1	eleq1d 1537	. . 3 |- (x = (/) -> ((aleph` x) e. On <-> (aleph` (/)) e. On))
3		fveq2 3715	. . . 4 |- (x = y -> (aleph` x) = (aleph` y))
4	3	eleq1d 1537	. . 3 |- (x = y -> ((aleph` x) e. On <-> (aleph` y) e. On))
5		fveq2 3715	. . . 4 |- (x = suc y -> (aleph` x) = (aleph` suc y))
6	5	eleq1d 1537	. . 3 |- (x = suc y -> ((aleph` x) e. On <-> (aleph` suc y) e. On))
7		fveq2 3715	. . . 4 |- (x = A -> (aleph` x) = (aleph` A))
8	7	eleq1d 1537	. . 3 |- (x = A -> ((aleph` x) e. On <-> (aleph` A) e. On))
9		aleph0 4843	. . . 4 |- (aleph` (/)) = om
10		omelon 4609	. . . 4 |- om e. On
11	9, 10	eqeltr 1541	. . 3 |- (aleph` (/)) e. On
12		ax-17 969	. . . . . . . . . 10 |- (w e. om -> A.z w e. om)
13		ax-17 969	. . . . . . . . . 10 |- (w e. y -> A.z w e. y)
14		ax-17 969	. . . . . . . . . 10 |- (w e. |^|{x e. On | (aleph` y) ~< x} -> A.z w e. |^|{x e. On | (aleph` y) ~< x})
15		df-aleph 4797	. . . . . . . . . 10 |- aleph = rec({<.z, y>. | y = |^|{x e. On | z ~< x}}, om)
16		breq1 2617	. . . . . . . . . . . 12 |- (z = (aleph` y) -> (z ~< x <-> (aleph` y) ~< x))
17	16	rabbisdv 1803	. . . . . . . . . . 11 |- (z = (aleph` y) -> {x e. On | z ~< x} = {x e. On | (aleph` y) ~< x})
18	17	inteqd 2533	. . . . . . . . . 10 |- (z = (aleph` y) -> |^|{x e. On | z ~< x} = |^|{x e. On | (aleph` y) ~< x})
19	12, 13, 14, 15, 18	rdgsucopab 3937	. . . . . . . . 9 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> (aleph` suc y) = |^|{x e. On | (aleph` y) ~< x})
20	19	eleq1d 1537	. . . . . . . 8 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> ((aleph` suc y) e. On <-> |^|{x e. On | (aleph` y) ~< x} e. On))
21		onintrab 3008	. . . . . . . 8 |- (|^|{x e. On | (aleph` y) ~< x} e. V <-> |^|{x e. On | (aleph` y) ~< x} e. On)
22	20, 21	syl6rbbr 538	. . . . . . 7 |- ((y e. On /\ |^|{x e. On | (aleph` y) ~< x} e. V) -> (|^|{x e. On | (aleph` y) ~< x} e. V <-> (aleph` suc y) e. On))
23	22	ex 373	. . . . . 6 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. V -> (|^|{x e. On | (aleph` y) ~< x} e. V <-> (aleph` suc y) e. On)))
24	23	ibd 593	. . . . 5 |- (y e. On -> (|^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On))
25	12, 13, 14, 15, 18	rdgsucopabn 3938	. . . . . 6 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) = (/))
26		0elon 3017	. . . . . 6 |- (/) e. On
27	25, 26	syl6eqel 1553	. . . . 5 |- (-. |^|{x e. On | (aleph` y) ~< x} e. V -> (aleph` suc y) e. On)
28	24, 27	pm2.61d1 128	. . . 4 |- (y e. On -> (aleph` suc y) e. On)
29	28	a1d 12	. . 3 |- (y e. On -> ((aleph` y) e. On -> (aleph` suc y) e. On))
30		visset 1809	. . . . . 6 |- x e. V
31		alephlim 4844	. . . . . 6 |- ((x e. V /\ Lim x) -> (aleph` x) = U_y e. x (aleph` y))
32	30, 31	mpan 694	. . . . 5 |- (Lim x -> (aleph` x) = U_y e. x (aleph` y))
33	32	eleq1d 1537	. . . 4 |- (Lim x -> ((aleph` x) e. On <-> U_y e. x (aleph` y) e. On))
34		fvex 3723	. . . . 5 |- (aleph` y) e. V
35	30, 34	iunon 3900	. . . 4 |- (A.y e. x (aleph` y) e. On -> U_y e. x (aleph` y) e. On)
36	33, 35	syl5bir 210	. . 3 |- (Lim x -> (A.y e. x (aleph` y) e. On -> (aleph` x) e. On))
37	2, 4, 6, 8, 11, 29, 36	tfinds 3156	. 2 |- (A e. On -> (aleph` A) e. On)
38		alephfnon 4842	. . . . . . 7 |- aleph Fn On
39		fndm 3579	. . . . . . 7 |- (aleph Fn On -> dom aleph = On)
40	38, 39	ax-mp 7	. . . . . 6 |- dom aleph = On
41	40	eleq2i 1535	. . . . 5 |- (A e. dom aleph <-> A e. On)
42	41	negbii 187	. . . 4 |- (-. A e. dom aleph <-> -. A e. On)
43		ndmfv 3736	. . . 4 |- (-. A e. dom aleph -> (aleph` A) = (/))
44	42, 43	sylbir 201	. . 3 |- (-. A e. On -> (aleph` A) = (/))
45	44, 26	syl6eqel 1553	. 2 |- (-. A e. On -> (aleph` A) e. On)
46	37, 45	pm2.61i 126	1 |- (aleph` A) e. On

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  A.wral 1642  {crab 1645  Vcvv 1807  (/)c0 2276  |^|cint 2528  U_ciun 2561   class class class wbr 2614  Oncon0 2943  Lim wlim 2944  suc csuc 2945  omcom 3126  dom cdm 3165   Fn wfn 3172  ` cfv 3177   ~< csdm 4356  alephcale 4794

This theorem is referenced by:  alephnbtwn 4848  alephnbtwn2 4849  alephordlem1 4852  alephordlem2 4853  alephordi 4854  alephord 4855  alephord2 4856  alephord3 4858  alephle 4864  cardaleph 4865  alephfp 4880  alephval2 4882

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-inf2 4605

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-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  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-fv 3193  df-rdg 3923  df-aleph 4797

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