Theorem onsubcum 14442

Description: An ordinal A is a subset of the cumulative hierarchy of sets at ordinal A.

Assertion

Ref	Expression
onsubcum	|- (A e. On -> A C_ (R1` A))

Proof of Theorem onsubcum

Step	Hyp	Ref	Expression
1		rankonid 5806	. . 3 |- (A e. On <-> (rank` A) = A)
2		fveq2 4681	. . . . . 6 |- (A = (rank` A) -> (R1` A) = (R1` (rank` A)))
3	2	eqcoms 1887	. . . . 5 |- ((rank` A) = A -> (R1` A) = (R1` (rank` A)))
4	3	sseq2d 2645	. . . 4 |- ((rank` A) = A -> (A C_ (R1` A) <-> A C_ (R1` (rank` A))))
5		r1rankid 5805	. . . 4 |- (A e. On -> A C_ (R1` (rank` A)))
6	4, 5	syl5bir 227	. . 3 |- ((rank` A) = A -> (A e. On -> A C_ (R1` A)))
7	1, 6	sylbi 216	. 2 |- (A e. On -> (A e. On -> A C_ (R1` A)))
8	7	pm2.43i 78	1 |- (A e. On -> A C_ (R1` A))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300   C_ wss 2593  Oncon0 3657  ` cfv 3998  R1cr1 5748  rankcrnk 5749

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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