Theorem cardmin 3666

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(s):   x,A

Proof of Theorem cardmin

Step	Hyp	Ref	Expression
1		numthcor 3601	. . . 4 |- (A e. B -> E.x e. On A ~< x)
2		onintrab2 2269	. . . 4 |- (E.x e. On A ~< x <-> |^|{x e. On | A ~< x} e. On)
3	1, 2	sylib 173	. . 3 |- (A e. B -> |^|{x e. On | A ~< x} e. On)
4		onelon 2223	. . . . . . . . . 10 |- ((|^|{x e. On | A ~< x} e. On /\ y e. |^|{x e. On | A ~< x}) -> y e. On)
5	4	exp 291	. . . . . . . . 9 |- (|^|{x e. On | A ~< x} e. On -> (y e. |^|{x e. On | A ~< x} -> y e. On))
6	3, 5	syl 12	. . . . . . . 8 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y e. On))
7		breq2 2066	. . . . . . . . . . . 12 |- (x = y -> (A ~< x <-> A ~< y))
8	7	elrab 1422	. . . . . . . . . . 11 |- (y e. {x e. On | A ~< x} <-> (y e. On /\ A ~< y))
9		ssrab 1556	. . . . . . . . . . . 12 |- {x e. On | A ~< x} (_ On
10		onnmin 2270	. . . . . . . . . . . 12 |- (({x e. On | A ~< x} (_ On /\ y e. {x e. On | A ~< x}) -> -. y e. |^|{x e. On | A ~< x})
11	9, 10	mpan 518	. . . . . . . . . . 11 |- (y e. {x e. On | A ~< x} -> -. y e. |^|{x e. On | A ~< x})
12	8, 11	sylbir 176	. . . . . . . . . 10 |- ((y e. On /\ A ~< y) -> -. y e. |^|{x e. On | A ~< x})
13	12	exp 291	. . . . . . . . 9 |- (y e. On -> (A ~< y -> -. y e. |^|{x e. On | A ~< x}))
14	13	con2d 83	. . . . . . . 8 |- (y e. On -> (y e. |^|{x e. On | A ~< x} -> -. A ~< y))
15	6, 14	syli 52	. . . . . . 7 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> -. A ~< y))
16		visset 1350	. . . . . . . 8 |- y e. V
17		domtri 3644	. . . . . . . 8 |- ((y e. V /\ A e. B) -> (y ~<_ A <-> -. A ~< y))
18	16, 17	mpan 518	. . . . . . 7 |- (A e. B -> (y ~<_ A <-> -. A ~< y))
19	15, 18	sylibrd 179	. . . . . 6 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y ~<_ A))
20		ax-17 925	. . . . . . . . . 10 |- (y e. A -> A.x y e. A)
21		ax-17 925	. . . . . . . . . 10 |- (y e. ~< -> A.x y e. ~< )
22		hbrab1 1310	. . . . . . . . . . 11 |- (y e. {x e. On | A ~< x} -> A.x y e. {x e. On | A ~< x})
23	22	hbint 1975	. . . . . . . . . 10 |- (y e. |^|{x e. On | A ~< x} -> A.x y e. |^|{x e. On | A ~< x})
24	20, 21, 23	hbbr 2095	. . . . . . . . 9 |- (A ~< |^|{x e. On | A ~< x} -> A.x A ~< |^|{x e. On | A ~< x})
25		breq2 2066	. . . . . . . . 9 |- (x = |^|{x e. On | A ~< x} -> (A ~< x <-> A ~< |^|{x e. On | A ~< x}))
26	24, 25	onminsb 2264	. . . . . . . 8 |- (E.x e. On A ~< x -> A ~< |^|{x e. On | A ~< x})
27	1, 26	syl 12	. . . . . . 7 |- (A e. B -> A ~< |^|{x e. On | A ~< x})
28	27	a1d 14	. . . . . 6 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> A ~< |^|{x e. On | A ~< x}))
29	19, 28	jcad 455	. . . . 5 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> (y ~<_ A /\ A ~< |^|{x e. On | A ~< x})))
30		domsdomtr 3374	. . . . 5 |- ((y ~<_ A /\ A ~< |^|{x e. On | A ~< x}) -> y ~< |^|{x e. On | A ~< x})
31	29, 30	syl6 23	. . . 4 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y ~< |^|{x e. On | A ~< x}))
32	31	r19.21aiv 1259	. . 3 |- (A e. B -> A.y e. |^| {x e. On | A ~< x}y ~< |^|{x e. On | A ~< x})
33	3, 32	jca 236	. 2 |- (A e. B -> (|^|{x e. On | A ~< x} e. On /\ A.y e. |^| {x e. On | A ~< x}y ~< |^|{x e. On | A ~< x}))
34		iscard 3659	. 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}))
35	33, 34	sylibr 175	1 |- (A e. B -> (card` |^|{x e. On | A ~< x}) = |^|{x e. On | A ~< x})

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  {crab 1204  Vcvv 1348   (_ wss 1487  |^|cint 1965   class class class wbr 2054  Oncon0 2199  ` cfv 2422   ~<_ cdom 3272   ~< csdm 3273  cardccrd 3620

This theorem is referenced by:  alephcard 3673

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623

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