Theorem rankel 4660

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 1473	. . . . 5 |- (rank` A) = (rank` A)
2		rankr1g 4655	. . . . 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 4651	. . . . . . . 8 |- (rank` A) e. On
6		r1suc 4632	. . . . . . . 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 1535	. . . . . 6 |- (B e. (R1` suc (rank` A)) <-> B e. P~(R1` (rank` A)))
9		rankel.1	. . . . . . 7 |- B e. V
10	9	elpw 2400	. . . . . 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 2059	. . . . 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 4651	. . . 4 |- (rank` B) e. On
17		ontri1 2976	. . . 4 |- (((rank` B) e. On /\ (rank` A) e. On) -> ((rank` B) (_ (rank` A) <-> -. (rank` A) e. (rank` B)))
18	16, 5, 17	mp2an 696	. . 3 |- ((rank` B) (_ (rank` A) <-> -. (rank` A) e. (rank` B))
19	16	onord 3090	. . . . 5 |- Ord (rank` B)
20	5	onord 3090	. . . . 5 |- Ord (rank` A)
21		ordsucsssuc 3069	. . . . 5 |- ((Ord (rank` B) /\ Ord (rank` A)) -> ((rank` B) (_ (rank` A) <-> suc (rank` B) (_ suc (rank` A)))
22	19, 20, 21	mp2an 696	. . . 4 |- ((rank` B) (_ (rank` A) <-> suc (rank` B) (_ suc (rank` A))
23	9	rankid 4652	. . . . 5 |- B e. (R1` suc (rank` B))
24	16	onsuc 3100	. . . . . . 7 |- suc (rank` B) e. On
25	5	onsuc 3100	. . . . . . 7 |- suc (rank` A) e. On
26		r1ord3 4637	. . . . . . 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 696	. . . . . 6 |- (suc (rank` B) (_ suc (rank` A) -> (R1` suc (rank` B)) (_ (R1` suc (rank` A)))
28	27	sseld 2063	. . . . 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 954   e. wcel 956  Vcvv 1807   (_ wss 2043  P~cpw 2397  Ord word 2942  Oncon0 2943  suc csuc 2945  ` cfv 3177  R1cr1 4621  rankcrnk 4622

This theorem is referenced by:  rankval3 4661  rankss 4668  rankuni2 4670  rankun 4671  rankuni 4678  rankval4 4682  rankc2 4686  rankxplim 4692

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-reg 4573  ax-inf2 4605

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-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  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-lim 2948  df-suc 2949  df-om 3127  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-fv 3193  df-rdg 3923  df-r1 4623  df-rank 4624

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