Theorem rankel 5791

Description: The membership relation is inherited by the rank function. Proposition 9.16 of [TakeutiZaring] p. 79.

Hypothesis

Ref	Expression
rankel.1	|- B e. _V

Assertion

Ref	Expression
rankel	|- (A e. B -> (rank` A) e. (rank` B))

Proof of Theorem rankel

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . 5 |- (rank` A) = (rank` A)
2		rankr1g 5786	. . . . 5 |- (A e. B -> ((rank` A) = (rank` A) <-> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A)))))
3	1, 2	mpbii 210	. . . 4 |- (A e. B -> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A))))
4	3	simplld 348	. . 3 |- (A e. B -> -. A e. (R1` (rank` A)))
5		rankon 5782	. . . . . . . 8 |- (rank` A) e. On
6		r1suc 5763	. . . . . . . 8 |- ((rank` A) e. On -> (R1` suc (rank` A)) = ~P(R1` (rank` A)))
7	5, 6	ax-mp 7	. . . . . . 7 |- (R1` suc (rank` A)) = ~P(R1` (rank` A))
8	7	eleq2i 1961	. . . . . 6 |- (B e. (R1` suc (rank` A)) <-> B e. ~P(R1` (rank` A)))
9		rankel.1	. . . . . . 7 |- B e. _V
10	9	elpw 3037	. . . . . 6 |- (B e. ~P(R1` (rank` A)) <-> B C_ (R1` (rank` A)))
11	8, 10	bitri 190	. . . . 5 |- (B e. (R1` suc (rank` A)) <-> B C_ (R1` (rank` A)))
12		ssel 2615	. . . . 5 |- (B C_ (R1` (rank` A)) -> (A e. B -> A e. (R1` (rank` A))))
13	11, 12	sylbi 216	. . . 4 |- (B e. (R1` suc (rank` A)) -> (A e. B -> A e. (R1` (rank` A))))
14	13	com12 14	. . 3 |- (A e. B -> (B e. (R1` suc (rank` A)) -> A e. (R1` (rank` A))))
15	4, 14	mtod 123	. 2 |- (A e. B -> -. B e. (R1` suc (rank` A)))
16		rankon 5782	. . . 4 |- (rank` B) e. On
17		ontri1 3695	. . . 4 |- (((rank` B) e. On /\ (rank` A) e. On) -> ((rank` B) C_ (rank` A) <-> -. (rank` A) e. (rank` B)))
18	16, 5, 17	mp2an 761	. . 3 |- ((rank` B) C_ (rank` A) <-> -. (rank` A) e. (rank` B))
19	16	onordi 3774	. . . . 5 |- Ord (rank` B)
20	5	onordi 3774	. . . . 5 |- Ord (rank` A)
21		ordsucsssuc 3904	. . . . 5 |- ((Ord (rank` B) /\ Ord (rank` A)) -> ((rank` B) C_ (rank` A) <-> suc (rank` B) C_ suc (rank` A)))
22	19, 20, 21	mp2an 761	. . . 4 |- ((rank` B) C_ (rank` A) <-> suc (rank` B) C_ suc (rank` A))
23	9	rankid 5783	. . . . 5 |- B e. (R1` suc (rank` B))
24	16	onsuci 3919	. . . . . . 7 |- suc (rank` B) e. On
25	5	onsuci 3919	. . . . . . 7 |- suc (rank` A) e. On
26		r1ord3 5768	. . . . . . 7 |- ((suc (rank` B) e. On /\ suc (rank` A) e. On) -> (suc (rank` B) C_ suc (rank` A) -> (R1` suc (rank` B)) C_ (R1` suc (rank` A))))
27	24, 25, 26	mp2an 761	. . . . . 6 |- (suc (rank` B) C_ suc (rank` A) -> (R1` suc (rank` B)) C_ (R1` suc (rank` A)))
28	27	sseld 2619	. . . . 5 |- (suc (rank` B) C_ suc (rank` A) -> (B e. (R1` suc (rank` B)) -> B e. (R1` suc (rank` A))))
29	23, 28	mpi 55	. . . 4 |- (suc (rank` B) C_ suc (rank` A) -> B e. (R1` suc (rank` A)))
30	22, 29	sylbi 216	. . 3 |- ((rank` B) C_ (rank` A) -> B e. (R1` suc (rank` A)))
31	18, 30	sylbir 218	. 2 |- (-. (rank` A) e. (rank` B) -> B e. (R1` suc (rank` A)))
32	15, 31	nsyl2 133	1 |- (A e. B -> (rank` A) e. (rank` B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  Ord word 3656  Oncon0 3657  suc csuc 3659  ` cfv 3998  R1cr1 5748  rankcrnk 5749

This theorem is referenced by:  rankval3 5792  rankss 5799  rankuni2 5801  rankun 5802  rankuni 5809  rankval4 5813  rankc2 5817  rankxplim 5823

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-reg 5695  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-rdg 5140  df-r1 5750  df-rank 5751

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