Theorem cardmin 4860

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 4786	. . . 4 |- (A e. B -> E.x e. On A ~< x)
2		onintrab2 3014	. . . 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 2972	. . . . . . . . . 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 2623	. . . . . . . . . . . 12 |- (x = y -> (A ~< x <-> A ~< y))
8	7	elrab 1905	. . . . . . . . . . 11 |- (y e. {x e. On | A ~< x} <-> (y e. On /\ A ~< y))
9		ssrab2 2131	. . . . . . . . . . . 12 |- {x e. On | A ~< x} (_ On
10		onnmin 3015	. . . . . . . . . . . 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 695	. . . . . . . . . . 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 1813	. . . . . . . 8 |- y e. V
17		domtri 4838	. . . . . . . 8 |- ((y e. V /\ A e. B) -> (y ~<_ A <-> -. A ~< y))
18	16, 17	mpan 695	. . . . . . 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-17 971	. . . . . . . . 9 |- (y e. A -> A.x y e. A)
21		ax-17 971	. . . . . . . . 9 |- (y e. ~< -> A.x y e. ~< )
22		hbrab1 1772	. . . . . . . . . 10 |- (y e. {x e. On | A ~< x} -> A.x y e. {x e. On | A ~< x})
23	22	hbint 2543	. . . . . . . . 9 |- (y e. |^|{x e. On | A ~< x} -> A.x y e. |^|{x e. On | A ~< x})
24	20, 21, 23	hbbr 2658	. . . . . . . 8 |- (A ~< |^|{x e. On | A ~< x} -> A.x A ~< |^|{x e. On | A ~< x})
25		breq2 2623	. . . . . . . 8 |- (x = |^|{x e. On | A ~< x} -> (A ~< x <-> A ~< |^|{x e. On | A ~< x}))
26	24, 25	onminsb 3009	. . . . . . 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 602	. . . . 5 |- (A e. B -> (y e. |^|{x e. On | A ~< x} -> (y ~<_ A /\ A ~< |^|{x e. On | A ~< x})))
29		domsdomtr 4476	. . . . 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 1713	. . 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 4853	. 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 956   e. wcel 958  A.wral 1645  E.wrex 1646  {crab 1648  Vcvv 1811   (_ wss 2047  |^|cint 2533   class class class wbr 2619  Oncon0 2948  ` cfv 3182   ~<_ cdom 4365   ~< csdm 4366  cardccrd 4813

This theorem is referenced by:  alephcard 4867

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-er 4261  df-en 4368  df-dom 4369  df-sdom 4370  df-card 4816

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