Theorem rankr1g 4675

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 3724	. . . 4 |- (x = A -> (rank` x) = (rank` A))
2	1	eqeq2d 1486	. . 3 |- (x = A -> (B = (rank` x) <-> B = (rank` A)))
3		eleq1 1534	. . . . 5 |- (x = A -> (x e. (R1` B) <-> A e. (R1` B)))
4	3	negbid 611	. . . 4 |- (x = A -> (-. x e. (R1` B) <-> -. A e. (R1` B)))
5		eleq1 1534	. . . 4 |- (x = A -> (x e. (R1` suc B) <-> A e. (R1` suc B)))
6	4, 5	anbi12d 628	. . 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 629	. 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 1813	. . 3 |- x e. V
9	8	rankr1 4674	. 2 |- (B = (rank` x) <-> (-. x e. (R1` B) /\ x e. (R1` suc B)))
10	7, 9	vtoclg 1847	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 956   e. wcel 958  suc csuc 2950  ` cfv 3182  R1cr1 4641  rankcrnk 4642

This theorem is referenced by:  rankel 4680  r1rankid 4694

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-r1 4643  df-rank 4644

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