Theorem rankr1g 4685

Description: A relationship between the rank function and the cumulative hierarchy of sets function R1. Proposition 9.15(2) of [TakeutiZaring] p. 79.

Assertion

Ref	Expression
rankr1g	|- (A e. C -> (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B))))

Proof of Theorem rankr1g

Step	Hyp	Ref	Expression
1		fveq2 3730	. . . 4 |- (x = A -> (rank` x) = (rank` A))
2	1	eqeq2d 1489	. . 3 |- (x = A -> (B = (rank` x) <-> B = (rank` A)))
3		eleq1 1537	. . . . 5 |- (x = A -> (x e. (R1` B) <-> A e. (R1` B)))
4	3	negbid 613	. . . 4 |- (x = A -> (-. x e. (R1` B) <-> -. A e. (R1` B)))
5		eleq1 1537	. . . 4 |- (x = A -> (x e. (R1` suc B) <-> A e. (R1` suc B)))
6	4, 5	anbi12d 630	. . 3 |- (x = A -> ((-. x e. (R1` B) /\ x e. (R1` suc B)) <-> (-. A e. (R1` B) /\ A e. (R1` suc B))))
7	2, 6	bibi12d 631	. 2 |- (x = A -> ((B = (rank` x) <-> (-. x e. (R1` B) /\ x e. (R1` suc B))) <-> (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B)))))
8		visset 1816	. . 3 |- x e. V
9	8	rankr1 4684	. 2 |- (B = (rank` x) <-> (-. x e. (R1` B) /\ x e. (R1` suc B)))
10	7, 9	vtoclg 1850	1 |- (A e. C -> (B = (rank` A) <-> (-. A e. (R1` B) /\ A e. (R1` suc B))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  suc csuc 2956  ` cfv 3188  R1cr1 4651  rankcrnk 4652

This theorem is referenced by:  rankel 4690  r1rankid 4704

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

Copyright terms: Public domain