Theorem ssrankr1 4656

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 (_ (rank` A) <-> -. A e. (R1` B)))

Proof of Theorem ssrankr1

Step	Hyp	Ref	Expression
1		eqid 1473	. . . . . 6 |- (rank` A) = (rank` A)
2		ssrankr1.1	. . . . . . 7 |- A e. V
3	2	rankr1 4654	. . . . . 6 |- ((rank` A) = (rank` A) <-> (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A))))
4	1, 3	mpbi 189	. . . . 5 |- (-. A e. (R1` (rank` A)) /\ A e. (R1` suc (rank` A)))
5	4	pm3.26i 320	. . . 4 |- -. A e. (R1` (rank` A))
6		rankon 4651	. . . . . . 7 |- (rank` A) e. On
7		r1ord3 4637	. . . . . . 7 |- ((B e. On /\ (rank` A) e. On) -> (B (_ (rank` A) -> (R1` B) (_ (R1` (rank` A))))
8	6, 7	mpan2 695	. . . . . 6 |- (B e. On -> (B (_ (rank` A) -> (R1` B) (_ (R1` (rank` A))))
9	8	imp 350	. . . . 5 |- ((B e. On /\ B (_ (rank` A)) -> (R1` B) (_ (R1` (rank` A)))
10	9	sseld 2063	. . . 4 |- ((B e. On /\ B (_ (rank` A)) -> (A e. (R1` B) -> A e. (R1` (rank` A))))
11	5, 10	mtoi 107	. . 3 |- ((B e. On /\ B (_ (rank` A)) -> -. A e. (R1` B))
12	11	ex 373	. 2 |- (B e. On -> (B (_ (rank` A) -> -. A e. (R1` B)))
13	2	rankr1lem 4653	. 2 |- (B e. On -> (-. A e. (R1` B) -> B (_ (rank` A)))
14	12, 13	impbid 515	1 |- (B e. On -> (B (_ (rank` A) <-> -. A e. (R1` B)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  Vcvv 1807   (_ wss 2043  Oncon0 2943  suc csuc 2945  ` cfv 3177  R1cr1 4621  rankcrnk 4622

This theorem is referenced by:  rankr1a 4657

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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