Theorem r1val3 5790

Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113.

Assertion

Ref	Expression
r1val3	|- (A e. On -> (R1` A) = U_x e. A ~P{y | (rank` y) e. x})

Distinct variable group:   x,y,A

Proof of Theorem r1val3

Step	Hyp	Ref	Expression
1		r1val1 5769	. 2 |- (A e. On -> (R1` A) = U_x e. A ~P(R1` x))
2		onelon 3683	. . . . 5 |- ((A e. On /\ x e. A) -> x e. On)
3		r1val2 5789	. . . . 5 |- (x e. On -> (R1` x) = {y | (rank` y) e. x})
4		pweq 3036	. . . . 5 |- ((R1` x) = {y | (rank` y) e. x} -> ~P(R1` x) = ~P{y | (rank` y) e. x})
5	2, 3, 4	3syl 24	. . . 4 |- ((A e. On /\ x e. A) -> ~P(R1` x) = ~P{y | (rank` y) e. x})
6	5	r19.21aiva 2176	. . 3 |- (A e. On -> A.x e. A ~P(R1` x) = ~P{y | (rank` y) e. x})
7		iuneq2 3273	. . 3 |- (A.x e. A ~P(R1` x) = ~P{y | (rank` y) e. x} -> U_x e. A ~P(R1` x) = U_x e. A ~P{y | (rank` y) e. x})
8	6, 7	syl 12	. 2 |- (A e. On -> U_x e. A ~P(R1` x) = U_x e. A ~P{y | (rank` y) e. x})
9	1, 8	eqtrd 1925	1 |- (A e. On -> (R1` A) = U_x e. A ~P{y | (rank` y) e. x})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  ~Pcpw 3032  U_ciun 3255  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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