Theorem rankel 4690

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 1478	. . . . 5 |- (rank` A) = (rank` A)
2		rankr1g 4685	. . . . 5 |- (A e. B -> ((rank` A) = (rank` A) <-> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A)))))
3	1, 2	mpbii 193	. . . 4 |- (A e. B -> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A))))
4	3	pm3.26d 321	. . 3 |- (A e. B -> -. A e. (R1` (rank` A)))
5		rankon 4681	. . . . . . . 8 |- (rank` A) e. On
6		r1suc 4662	. . . . . . . 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 1541	. . . . . 6 |- (B e. (R1` suc (rank` A)) <-> B e. P~(R1` (rank` A)))
9		rankel.1	. . . . . . 7 |- B e. V
10	9	elpw 2408	. . . . . 6 |- (B e. P~(R1` (rank` A)) <-> B (_ (R1` (rank` A)))
11	8, 10	bitr 173	. . . . 5 |- (B e. (R1` suc (rank` A)) <-> B (_ (R1` (rank` A)))
12		ssel 2066	. . . . 5 |- (B (_ (R1` (rank` A)) -> (A e. B -> A e. (R1` (rank` A))))
13	11, 12	sylbi 199	. . . 4 |- (B e. (R1` suc (rank` A)) -> (A e. B -> A e. (R1` (rank` A))))
14	13	com12 11	. . 3 |- (A e. B -> (B e. (R1` suc (rank` A)) -> A e. (R1` (rank` A))))
15	4, 14	mtod 108	. 2 |- (A e. B -> -. B e. (R1` suc (rank` A)))
16		rankon 4681	. . . 4 |- (rank` B) e. On
17		ontri1 2987	. . . 4 |- (((rank` B) e. On /\ (rank` A) e. On) -> ((rank` B) (_ (rank` A) <-> -. (rank` A) e. (rank` B)))
18	16, 5, 17	mp2an 699	. . 3 |- ((rank` B) (_ (rank` A) <-> -. (rank` A) e. (rank` B))
19	16	onord 3101	. . . . 5 |- Ord (rank` B)
20	5	onord 3101	. . . . 5 |- Ord (rank` A)
21		ordsucsssuc 3080	. . . . 5 |- ((Ord (rank` B) /\ Ord (rank` A)) -> ((rank` B) (_ (rank` A) <-> suc (rank` B) (_ suc (rank` A)))
22	19, 20, 21	mp2an 699	. . . 4 |- ((rank` B) (_ (rank` A) <-> suc (rank` B) (_ suc (rank` A))
23	9	rankid 4682	. . . . 5 |- B e. (R1` suc (rank` B))
24	16	onsuc 3111	. . . . . . 7 |- suc (rank` B) e. On
25	5	onsuc 3111	. . . . . . 7 |- suc (rank` A) e. On
26		r1ord3 4667	. . . . . . 7 |- ((suc (rank` B) e. On /\ suc (rank` A) e. On) -> (suc (rank` B) (_ suc (rank` A) -> (R1` suc (rank` B)) (_ (R1` suc (rank` A))))
27	24, 25, 26	mp2an 699	. . . . . 6 |- (suc (rank` B) (_ suc (rank` A) -> (R1` suc (rank` B)) (_ (R1` suc (rank` A)))
28	27	sseld 2070	. . . . 5 |- (suc (rank` B) (_ suc (rank` A) -> (B e. (R1` suc (rank` B)) -> B e. (R1` suc (rank` A))))
29	23, 28	mpi 44	. . . 4 |- (suc (rank` B) (_ suc (rank` A) -> B e. (R1` suc (rank` A)))
30	22, 29	sylbi 199	. . 3 |- ((rank` B) (_ (rank` A) -> B e. (R1` suc (rank` A)))
31	18, 30	sylbir 201	. 2 |- (-. (rank` A) e. (rank` B) -> B e. (R1` suc (rank` A)))
32	15, 31	nsyl2 118	1 |- (A e. B -> (rank` A) e. (rank` B))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814   (_ wss 2050  P~cpw 2405  Ord word 2953  Oncon0 2954  suc csuc 2956  ` cfv 3188  R1cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankval3 4691  rankss 4698  rankuni2 4700  rankun 4701  rankuni 4708  rankval4 4712  rankc2 4716  rankxplim 4722

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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