Theorem ssrankr1 5787

Description: A relationship between an ordinal number less than or equal to a rank, and the cumulative hierarchy of sets R1. Proposition 9.15(3) of [TakeutiZaring] p. 79.

Hypothesis

Ref	Expression
ssrankr1.1	|- A e. _V

Assertion

Ref	Expression
ssrankr1	|- (B e. On -> (B C_ (rank` A) <-> -. A e. (R1` B)))

Proof of Theorem ssrankr1

Step	Hyp	Ref	Expression
1		eqid 1884	. . . . . 6 |- (rank` A) = (rank` A)
2		ssrankr1.1	. . . . . . 7 |- A e. _V
3	2	rankr1 5785	. . . . . 6 |- ((rank` A) = (rank` A) <-> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A))))
4	1, 3	mpbi 206	. . . . 5 |- (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A)))
5	4	simpli 347	. . . 4 |- -. A e. (R1` (rank` A))
6		rankon 5782	. . . . . . 7 |- (rank` A) e. On
7		r1ord3 5768	. . . . . . 7 |- ((B e. On /\ (rank` A) e. On) -> (B C_ (rank` A) -> (R1` B) C_ (R1` (rank` A))))
8	6, 7	mpan2 760	. . . . . 6 |- (B e. On -> (B C_ (rank` A) -> (R1` B) C_ (R1` (rank` A))))
9	8	imp 377	. . . . 5 |- ((B e. On /\ B C_ (rank` A)) -> (R1` B) C_ (R1` (rank` A)))
10	9	sseld 2619	. . . 4 |- ((B e. On /\ B C_ (rank` A)) -> (A e. (R1` B) -> A e. (R1` (rank` A))))
11	5, 10	mtoi 122	. . 3 |- ((B e. On /\ B C_ (rank` A)) -> -. A e. (R1` B))
12	11	ex 402	. 2 |- (B e. On -> (B C_ (rank` A) -> -. A e. (R1` B)))
13	2	rankr1lem 5784	. 2 |- (B e. On -> (-. A e. (R1` B) -> B C_ (rank` A)))
14	12, 13	impbid 574	1 |- (B e. On -> (B C_ (rank` A) <-> -. A e. (R1` B)))

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

This theorem is referenced by:  rankr1a 5788

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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