Theorem cardmin 4843

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		numthcor 4769	. . . 4 |- (A e. B -> E.x e. On A ~< x)
2		onintrab2 3010	. . . 4 |- (E.x e. On A ~< x <-> |^|{x e. On | A ~< x} e. On)
3	1, 2	sylib 198	. . 3 |- (A e. B -> |^|{x e. On | A ~< x} e. On)
4		onelon 2968	. . . . . . . . . 10 |- ((|^|{x e. On | A ~< x} e. On /\ y e. |^|{x e. On | A ~< x}) -> y e. On)
5	4	ex 373	. . . . . . . . 9 |- (|^|{x e. On | A ~< x} e. On -> (y e. |^|{x e. On | A ~< x} -> y e. On))
6	3, 5	syl 10	. . . . . . . 8 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y e. On))
7		breq2 2619	. . . . . . . . . . . 12 |- (x = y -> (A ~< x <-> A ~< y))
8	7	elrab 1902	. . . . . . . . . . 11 |- (y e. {x e. On | A ~< x} <-> (y e. On /\ A ~< y))
9		ssrab2 2128	. . . . . . . . . . . 12 |- {x e. On | A ~< x} (_ On
10		onnmin 3011	. . . . . . . . . . . 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 694	. . . . . . . . . . 11 |- (y e. {x e. On | A ~< x} -> -. y e. |^|{x e. On | A ~< x})
12	8, 11	sylbir 201	. . . . . . . . . 10 |- ((y e. On /\ A ~< y) -> -. y e. |^|{x e. On | A ~< x})
13	12	ex 373	. . . . . . . . 9 |- (y e. On -> (A ~< y -> -. y e. |^|{x e. On | A ~< x}))
14	13	con2d 91	. . . . . . . 8 |- (y e. On -> (y e. |^|{x e. On | A ~< x} -> -. A ~< y))
15	6, 14	syli 54	. . . . . . 7 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> -. A ~< y))
16		visset 1810	. . . . . . . 8 |- y e. V
17		domtri 4821	. . . . . . . 8 |- ((y e. V /\ A e. B) -> (y ~<_ A <-> -. A ~< y))
18	16, 17	mpan 694	. . . . . . 7 |- (A e. B -> (y ~<_ A <-> -. A ~< y))
19	15, 18	sylibrd 204	. . . . . 6 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y ~<_ A))
20		ax-17 970	. . . . . . . . 9 |- (y e. A -> A.x y e. A)
21		ax-17 970	. . . . . . . . 9 |- (y e. ~< -> A.x y e. ~< )
22		hbrab1 1770	. . . . . . . . . 10 |- (y e. {x e. On | A ~< x} -> A.x y e. {x e. On | A ~< x})
23	22	hbint 2539	. . . . . . . . 9 |- (y e. |^|{x e. On | A ~< x} -> A.x y e. |^|{x e. On | A ~< x})
24	20, 21, 23	hbbr 2654	. . . . . . . 8 |- (A ~< |^|{x e. On | A ~< x} -> A.x A ~< |^|{x e. On | A ~< x})
25		breq2 2619	. . . . . . . 8 |- (x = |^|{x e. On | A ~< x} -> (A ~< x <-> A ~< |^|{x e. On | A ~< x}))
26	24, 25	onminsb 3005	. . . . . . 7 |- (E.x e. On A ~< x -> A ~< |^|{x e. On | A ~< x})
27	1, 26	syl 10	. . . . . 6 |- (A e. B -> A ~< |^|{x e. On | A ~< x})
28	19, 27	jctird 601	. . . . 5 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> (y ~<_ A /\ A ~< |^|{x e. On | A ~< x})))
29		domsdomtr 4465	. . . . 5 |- ((y ~<_ A /\ A ~< |^|{x e. On | A ~< x}) -> y ~< |^|{x e. On | A ~< x})
30	28, 29	syl6 22	. . . 4 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> y ~< |^|{x e. On | A ~< x}))
31	30	r19.21aiv 1711	. . 3 |- (A e. B -> A.y e. |^|{x e. On | A ~< x}y ~< |^|{x e. On | A ~< x})
32	3, 31	jca 288	. 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}))
33		iscard 4836	. 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}))
34	32, 33	sylibr 200	1 |- (A e. B -> (card` |^|{x e. On | A ~< x}) = |^|{x e. On | A ~< x})

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643  E.wrex 1644  {crab 1646  Vcvv 1808   (_ wss 2044  |^|cint 2529   class class class wbr 2615  Oncon0 2944  ` cfv 3178   ~<_ cdom 4358   ~< csdm 4359  cardccrd 4796

This theorem is referenced by:  alephcard 4850

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194  df-er 4254  df-en 4360  df-dom 4361  df-sdom 4362  df-card 4799

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