Theorem rankeq0 4696

Description: A set is empty iff its rank is empty.

Hypothesis

Ref	Expression
rankeq0.1	|- A e. V

Assertion

Ref	Expression
rankeq0	|- (A = (/) <-> (rank` A) = (/))

Proof of Theorem rankeq0

Step	Hyp	Ref	Expression
1		fveq2 3724	. . 3 |- (A = (/) -> (rank` A) = (rank` (/)))
2		0elon 3022	. . . 4 |- (/) e. On
3		rankonid 4695	. . . 4 |- ((/) e. On <-> (rank` (/)) = (/))
4	2, 3	mpbi 189	. . 3 |- (rank` (/)) = (/)
5	1, 4	syl6eq 1523	. 2 |- (A = (/) -> (rank` A) = (/))
6		rankeq0.1	. . . . 5 |- A e. V
7		rankval2 4670	. . . . 5 |- (A e. V -> (rank` A) = |^|{x e. On | A (_ (R1` x)})
8	6, 7	ax-mp 7	. . . 4 |- (rank` A) = |^|{x e. On | A (_ (R1` x)}
9	8	eqeq1i 1482	. . 3 |- ((rank` A) = (/) <-> |^|{x e. On | A (_ (R1` x)} = (/))
10		ssrab2 2131	. . . . . 6 |- {x e. On | A (_ (R1` x)} (_ On
11		onint0 3007	. . . . . 6 |- ({x e. On | A (_ (R1` x)} (_ On -> (|^|{x e. On | A (_ (R1` x)} = (/) <-> (/) e. {x e. On | A (_ (R1` x)}))
12	10, 11	ax-mp 7	. . . . 5 |- (|^|{x e. On | A (_ (R1` x)} = (/) <-> (/) e. {x e. On | A (_ (R1` x)})
13		fveq2 3724	. . . . . . 7 |- (x = (/) -> (R1` x) = (R1` (/)))
14	13	sseq2d 2089	. . . . . 6 |- (x = (/) -> (A (_ (R1` x) <-> A (_ (R1` (/))))
15	14	elrab 1905	. . . . 5 |- ((/) e. {x e. On | A (_ (R1` x)} <-> ((/) e. On /\ A (_ (R1` (/))))
16	2	biantrur 725	. . . . . 6 |- (A (_ (R1` (/)) <-> ((/) e. On /\ A (_ (R1` (/))))
17		r10 4651	. . . . . . 7 |- (R1` (/)) = (/)
18	17	sseq2i 2086	. . . . . 6 |- (A (_ (R1` (/)) <-> A (_ (/))
19	16, 18	bitr3 175	. . . . 5 |- (((/) e. On /\ A (_ (R1` (/))) <-> A (_ (/))
20	12, 15, 19	3bitr 177	. . . 4 |- (|^|{x e. On | A (_ (R1` x)} = (/) <-> A (_ (/))
21		ss0 2303	. . . 4 |- (A (_ (/) -> A = (/))
22	20, 21	sylbi 199	. . 3 |- (|^|{x e. On | A (_ (R1` x)} = (/) -> A = (/))
23	9, 22	sylbi 199	. 2 |- ((rank` A) = (/) -> A = (/))
24	5, 23	impbi 157	1 |- (A = (/) <-> (rank` A) = (/))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648  Vcvv 1811   (_ wss 2047  (/)c0 2280  |^|cint 2533  Oncon0 2948  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  rankxplim2 4713  rankxplim3 4714  rankxpsuc 4715

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-reg 4593  ax-inf2 4625

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-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  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-lim 2953  df-suc 2954  df-om 3132  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-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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