Theorem rankel 3524

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		cleqid 1102	. . . . 5 |- (rank` A) = (rank` A)
2		rankr1g 3519	. . . . 5 |- (A e. B -> ((rank` A) = (rank` A) <-> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A)))))
3	1, 2	mpbii 168	. . . 4 |- (A e. B -> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A))))
4	3	pm3.26d 258	. . 3 |- (A e. B -> -. A e. (R1` (rank` A)))
5		rankon 3515	. . . . . . . 8 |- (rank` A) e. On
6		r1suc 3496	. . . . . . . 8 |- ((rank` A) e. On -> (R1` suc (rank` A)) = P~(R1` (rank` A)))
7	5, 6	ax-mp 6	. . . . . . 7 |- (R1` suc (rank` A)) = P~(R1` (rank` A))
8	7	eleq2i 1153	. . . . . 6 |- (B e. (R1` suc (rank` A)) <-> B e. P~(R1` (rank` A)))
9		rankel.1	. . . . . . 7 |- B e. V
10	9	elpw 1801	. . . . . 6 |- (B e. P~(R1` (rank` A)) <-> B (_ (R1` (rank` A)))
11	8, 10	bitr 151	. . . . 5 |- (B e. (R1` suc (rank` A)) <-> B (_ (R1` (rank` A)))
12		ssel 1502	. . . . 5 |- (B (_ (R1` (rank` A)) -> (A e. B -> A e. (R1` (rank` A))))
13	11, 12	sylbi 174	. . . 4 |- (B e. (R1` suc (rank` A)) -> (A e. B -> A e. (R1` (rank` A))))
14	13	com12 13	. . 3 |- (A e. B -> (B e. (R1` suc (rank` A)) -> A e. (R1` (rank` A))))
15	4, 14	mtod 95	. 2 |- (A e. B -> -. B e. (R1` suc (rank` A)))
16		rankon 3515	. . . 4 |- (rank` B) e. On
17		ontri1 2232	. . . 4 |- (((rank` B) e. On /\ (rank` A) e. On) -> ((rank` B) (_ (rank` A) <-> -. (rank` A) e. (rank` B)))
18	16, 5, 17	mp2an 520	. . 3 |- ((rank` B) (_ (rank` A) <-> -. (rank` A) e. (rank` B))
19	16	onord 2343	. . . . 5 |- Ord (rank` B)
20	5	onord 2343	. . . . 5 |- Ord (rank` A)
21		ordsucsssuc 2325	. . . . 5 |- ((Ord (rank` B) /\ Ord (rank` A)) -> ((rank` B) (_ (rank` A) <-> suc (rank` B) (_ suc (rank` A)))
22	19, 20, 21	mp2an 520	. . . 4 |- ((rank` B) (_ (rank` A) <-> suc (rank` B) (_ suc (rank` A))
23	9	rankid 3516	. . . . 5 |- B e. (R1` suc (rank` B))
24	16	onsuc 2353	. . . . . . 7 |- suc (rank` B) e. On
25	5	onsuc 2353	. . . . . . 7 |- suc (rank` A) e. On
26		r1ord3 3501	. . . . . . 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 520	. . . . . 6 |- (suc (rank` B) (_ suc (rank` A) -> (R1` suc (rank` B)) (_ (R1` suc (rank` A)))
28	27	sseld 1506	. . . . 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 174	. . 3 |- ((rank` B) (_ (rank` A) -> B e. (R1` suc (rank` A)))
31	18, 30	sylbir 176	. 2 |- (-. (rank` A) e. (rank` B) -> B e. (R1` suc (rank` A)))
32	15, 31	nsyl2 103	1 |- (A e. B -> (rank` A) e. (rank` B))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348   (_ wss 1487  P~cpw 1798  Ord word 2198  Oncon0 2199  suc csuc 2201  ` cfv 2422  R1cr1 3485  rankcrnk 3486

This theorem is referenced by:  rankval3 3525  rankss 3531  rankuni 3533  rankuniss 3534  rankun 3535  ranklon 3540

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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