Theorem cardmin 6012

Description: The smallest ordinal that strictly dominates a set is a cardinal.

Assertion

Ref	Expression
cardmin	|- (A e. B -> (card` |^|{x e. On | A ~< x}) = |^|{x e. On | A ~< x})

Distinct variable group:   x,A

Proof of Theorem cardmin

Step	Hyp	Ref	Expression
1		iscard 6005	. 2 |- ((card` |^|{x e. On | A ~< x}) = |^|{x e. On | A ~< x} <-> (|^|{x e. On | A ~< x} e. On /\ A.y e. |^|{x e. On | A ~< x}y ~< |^|{x e. On | A ~< x}))
2		numthcor 5948	. . 3 |- (A e. B -> E.x e. On A ~< x)
3		onintrab2 3883	. . 3 |- (E.x e. On A ~< x <-> |^|{x e. On | A ~< x} e. On)
4	2, 3	sylib 215	. 2 |- (A e. B -> |^|{x e. On | A ~< x} e. On)
5		onelon 3683	. . . . . . . . 9 |- ((|^|{x e. On | A ~< x} e. On /\ y e. |^|{x e. On | A ~< x}) -> y e. On)
6	5	ex 402	. . . . . . . 8 |- (|^|{x e. On | A ~< x} e. On -> (y e. |^|{x e. On | A ~< x} -> y e. On))
7	4, 6	syl 12	. . . . . . 7 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y e. On))
8		breq2 3342	. . . . . . . . . . 11 |- (x = y -> (A ~< x <-> A ~< y))
9	8	elrab 2414	. . . . . . . . . 10 |- (y e. {x e. On | A ~< x} <-> (y e. On /\ A ~< y))
10		ssrab2 2692	. . . . . . . . . . 11 |- {x e. On | A ~< x} C_ On
11		onnmin 3884	. . . . . . . . . . 11 |- (({x e. On | A ~< x} C_ On /\ y e. {x e. On | A ~< x}) -> -. y e. |^|{x e. On | A ~< x})
12	10, 11	mpan 759	. . . . . . . . . 10 |- (y e. {x e. On | A ~< x} -> -. y e. |^|{x e. On | A ~< x})
13	9, 12	sylbir 218	. . . . . . . . 9 |- ((y e. On /\ A ~< y) -> -. y e. |^|{x e. On | A ~< x})
14	13	ex 402	. . . . . . . 8 |- (y e. On -> (A ~< y -> -. y e. |^|{x e. On | A ~< x}))
15	14	con2d 107	. . . . . . 7 |- (y e. On -> (y e. |^|{x e. On | A ~< x} -> -. A ~< y))
16	7, 15	syli 65	. . . . . 6 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> -. A ~< y))
17		visset 2295	. . . . . . 7 |- y e. _V
18		domtri 5989	. . . . . . 7 |- ((y e. _V /\ A e. B) -> (y ~<_ A <-> -. A ~< y))
19	17, 18	mpan 759	. . . . . 6 |- (A e. B -> (y ~<_ A <-> -. A ~< y))
20	16, 19	sylibrd 221	. . . . 5 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y ~<_ A))
21		ax-17 1317	. . . . . . . 8 |- (y e. A -> A.x y e. A)
22		ax-17 1317	. . . . . . . 8 |- (y e. ~< -> A.x y e. ~< )
23		hbrab1 2257	. . . . . . . . 9 |- (y e. {x e. On | A ~< x} -> A.x y e. {x e. On | A ~< x})
24	23	hbint 3225	. . . . . . . 8 |- (y e. |^|{x e. On | A ~< x} -> A.x y e. |^|{x e. On | A ~< x})
25	21, 22, 24	hbbr 3381	. . . . . . 7 |- (A ~< |^|{x e. On | A ~< x} -> A.x A ~< |^|{x e. On | A ~< x})
26		breq2 3342	. . . . . . 7 |- (x = |^|{x e. On | A ~< x} -> (A ~< x <-> A ~< |^|{x e. On | A ~< x}))
27	25, 26	onminsb 3879	. . . . . 6 |- (E.x e. On A ~< x -> A ~< |^|{x e. On | A ~< x})
28	2, 27	syl 12	. . . . 5 |- (A e. B -> A ~< |^|{x e. On | A ~< x})
29	20, 28	jctird 663	. . . 4 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> (y ~<_ A /\ A ~< |^|{x e. On | A ~< x})))
30		domsdomtr 5539	. . . 4 |- ((y ~<_ A /\ A ~< |^|{x e. On | A ~< x}) -> y ~< |^|{x e. On | A ~< x})
31	29, 30	syl6 25	. . 3 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y ~< |^|{x e. On | A ~< x}))
32	31	r19.21aiv 2175	. 2 |- (A e. B -> A.y e. |^|{x e. On | A ~< x}y ~< |^|{x e. On | A ~< x})
33	1, 4, 32	sylanbrc 527	1 |- (A e. B -> (card` |^|{x e. On | A ~< x}) = |^|{x e. On | A ~< x})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  {crab 2108  _Vcvv 2292   C_ wss 2593  |^|cint 3214   class class class wbr 3338  Oncon0 3657  ` cfv 3998   ~<_ cdom 5424   ~< csdm 5425  cardccrd 5859

This theorem is referenced by:  alephcard 6015

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-card 5862

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