Theorem r1rankid 4704

Description: Any set is a subset of the hierarchy of its rank.

Assertion

Ref	Expression
r1rankid	|- (A e. B -> A (_ (R1` (rank` A)))

Proof of Theorem r1rankid

Step	Hyp	Ref	Expression
1		eqid 1478	. . . . 5 |- (rank` A) = (rank` A)
2		rankr1g 4685	. . . . 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	pm3.27d 325	. . 3 |- (A e. B -> A e. (R1` suc (rank` A)))
5		rankon 4681	. . . 4 |- (rank` A) e. On
6		r1suc 4662	. . . 4 |- ((rank` A) e. On -> (R1` suc (rank` A)) = P~(R1` (rank` A)))
7	5, 6	ax-mp 7	. . 3 |- (R1` suc (rank` A)) = P~(R1` (rank` A))
8	4, 7	syl6eleq 1561	. 2 |- (A e. B -> A e. P~(R1` (rank` A)))
9		elpwg 2409	. 2 |- (A e. B -> (A e. P~(R1` (rank` A)) <-> A (_ (R1` (rank` A))))
10	8, 9	mpbid 195	1 |- (A e. B -> A (_ (R1` (rank` A)))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960   (_ wss 2050  P~cpw 2405  Oncon0 2954  suc csuc 2956  ` cfv 3188  R1cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankr1id 4707  rankr1b 4709

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-rdg 3938  df-r1 4653  df-rank 4654

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