Theorem cfeq0 4894

Description: Only the ordinal zero has cofinality zero.

Assertion

Ref	Expression
cfeq0	|- (A e. On -> ((cf` A) = (/) <-> A = (/)))

Proof of Theorem cfeq0

Step	Hyp	Ref	Expression
1		cfval 4886	. . . 4 |- (A e. On -> (cf` A) = |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
2	1	eqeq1d 1480	. . 3 |- (A e. On -> ((cf` A) = (/) <-> |^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} = (/)))
3		visset 1809	. . . . . . . . 9 |- v e. V
4		eqeq1 1478	. . . . . . . . . . 11 |- (x = v -> (x = (card` y) <-> v = (card` y)))
5	4	anbi1d 616	. . . . . . . . . 10 |- (x = v -> ((x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> (v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
6	5	exbidv 1277	. . . . . . . . 9 |- (x = v -> (E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
7	3, 6	elab 1893	. . . . . . . 8 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} <-> E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
8		fveq2 3715	. . . . . . . . . . . 12 |- (v = (card` y) -> (card` v) = (card` (card` y)))
9		cardidm 4829	. . . . . . . . . . . 12 |- (card` (card` y)) = (card` y)
10	8, 9	syl6eq 1520	. . . . . . . . . . 11 |- (v = (card` y) -> (card` v) = (card` y))
11		eqeq2 1481	. . . . . . . . . . 11 |- (v = (card` y) -> ((card` v) = v <-> (card` v) = (card` y)))
12	10, 11	mpbird 196	. . . . . . . . . 10 |- (v = (card` y) -> (card` v) = v)
13	12	adantr 389	. . . . . . . . 9 |- ((v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> (card` v) = v)
14	13	19.23aiv 1293	. . . . . . . 8 |- (E.y(v = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> (card` v) = v)
15	7, 14	sylbi 199	. . . . . . 7 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> (card` v) = v)
16		cardon 4807	. . . . . . 7 |- (card` v) e. On
17	15, 16	syl6eqelr 1554	. . . . . 6 |- (v e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> v e. On)
18	17	ssriv 2065	. . . . 5 |- {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ On
19		onint0 3002	. . . . 5 |- ({x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} (_ On -> (|^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} = (/) <-> (/) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))}))
20	18, 19	ax-mp 7	. . . 4 |- (|^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} = (/) <-> (/) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))})
21		0ex 2706	. . . . . 6 |- (/) e. V
22		eqeq1 1478	. . . . . . . 8 |- (x = (/) -> (x = (card` y) <-> (/) = (card` y)))
23	22	anbi1d 616	. . . . . . 7 |- (x = (/) -> ((x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> ((/) = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
24	23	exbidv 1277	. . . . . 6 |- (x = (/) -> (E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) <-> E.y((/) = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))))
25	21, 24	elab 1893	. . . . 5 |- ((/) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} <-> E.y((/) = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)))
26		sseq1 2078	. . . . . . . . . 10 |- (y = (/) -> (y (_ A <-> (/) (_ A))
27		rexeq1 1784	. . . . . . . . . . 11 |- (y = (/) -> (E.w e. y z (_ w <-> E.w e. (/) z (_ w))
28	27	ralbidv 1660	. . . . . . . . . 10 |- (y = (/) -> (A.z e. A E.w e. y z (_ w <-> A.z e. A E.w e. (/) z (_ w))
29	26, 28	anbi12d 627	. . . . . . . . 9 |- (y = (/) -> ((y (_ A /\ A.z e. A E.w e. y z (_ w) <-> ((/) (_ A /\ A.z e. A E.w e. (/) z (_ w)))
30	29	biimpa 416	. . . . . . . 8 |- ((y = (/) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> ((/) (_ A /\ A.z e. A E.w e. (/) z (_ w))
31		eqcom 1474	. . . . . . . . 9 |- ((/) = (card` y) <-> (card` y) = (/))
32		visset 1809	. . . . . . . . . 10 |- y e. V
33		cardeq0 4812	. . . . . . . . . 10 |- (y e. V -> ((card` y) = (/) <-> y = (/)))
34	32, 33	ax-mp 7	. . . . . . . . 9 |- ((card` y) = (/) <-> y = (/))
35	31, 34	bitr 173	. . . . . . . 8 |- ((/) = (card` y) <-> y = (/))
36	30, 35	sylanb 449	. . . . . . 7 |- (((/) = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> ((/) (_ A /\ A.z e. A E.w e. (/) z (_ w))
37		rex0 2287	. . . . . . . . . . . . 13 |- -. E.w e. (/) z (_ w
38	37	a1i 8	. . . . . . . . . . . 12 |- (z e. A -> -. E.w e. (/) z (_ w)
39	38	rgen 1695	. . . . . . . . . . 11 |- A.z e. A -. E.w e. (/) z (_ w
40		r19.2z 2343	. . . . . . . . . . 11 |- ((A =/= (/) /\ A.z e. A -. E.w e. (/) z (_ w) -> E.z e. A -. E.w e. (/) z (_ w)
41	39, 40	mpan2 695	. . . . . . . . . 10 |- (A =/= (/) -> E.z e. A -. E.w e. (/) z (_ w)
42		rexnal 1651	. . . . . . . . . 10 |- (E.z e. A -. E.w e. (/) z (_ w <-> -. A.z e. A E.w e. (/) z (_ w)
43	41, 42	sylib 198	. . . . . . . . 9 |- (A =/= (/) -> -. A.z e. A E.w e. (/) z (_ w)
44	43	necon4ai 1621	. . . . . . . 8 |- (A.z e. A E.w e. (/) z (_ w -> A = (/))
45	44	adantl 388	. . . . . . 7 |- (((/) (_ A /\ A.z e. A E.w e. (/) z (_ w) -> A = (/))
46	36, 45	syl 10	. . . . . 6 |- (((/) = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> A = (/))
47	46	19.23aiv 1293	. . . . 5 |- (E.y((/) = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w)) -> A = (/))
48	25, 47	sylbi 199	. . . 4 |- ((/) e. {x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} -> A = (/))
49	20, 48	sylbi 199	. . 3 |- (|^|{x | E.y(x = (card` y) /\ (y (_ A /\ A.z e. A E.w e. y z (_ w))} = (/) -> A = (/))
50	2, 49	syl6bi 214	. 2 |- (A e. On -> ((cf` A) = (/) -> A = (/)))
51		fveq2 3715	. . 3 |- (A = (/) -> (cf` A) = (cf` (/)))
52		cf0 4890	. . 3 |- (cf` (/)) = (/)
53	51, 52	syl6eq 1520	. 2 |- (A = (/) -> (cf` A) = (/))
54	50, 53	impbid1 516	1 |- (A e. On -> ((cf` A) = (/) <-> A = (/)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  {cab 1461   =/= wne 1582  A.wral 1642  E.wrex 1643  Vcvv 1807   (_ wss 2043  (/)c0 2276  |^|cint 2528  Oncon0 2943  ` cfv 3177  cardccrd 4793  cfccf 4795

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-ac 4724

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-reu 1648  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-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-suc 2949  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-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-er 4251  df-en 4357  df-card 4796  df-cf 4798

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