Theorem rankwflem 5776

Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 5775 is useful in proofs of theorems about the rank function.

Assertion

Ref	Expression
rankwflem	|- (A e. B -> E.x e. On A e. (R1` suc x))

Distinct variable group:   x,A

Proof of Theorem rankwflem

Step	Hyp	Ref	Expression
1		tz9.13g 5775	. 2 |- (A e. B -> E.x e. On A e. (R1` x))
2		suceloni 3894	. . . . 5 |- (x e. On -> suc x e. On)
3		visset 2295	. . . . . . 7 |- x e. _V
4	3	sucid 3744	. . . . . 6 |- x e. suc x
5		r1ord2 5767	. . . . . 6 |- (suc x e. On -> (x e. suc x -> (R1` x) C_ (R1` suc x)))
6	4, 5	mpi 55	. . . . 5 |- (suc x e. On -> (R1` x) C_ (R1` suc x))
7	2, 6	syl 12	. . . 4 |- (x e. On -> (R1` x) C_ (R1` suc x))
8	7	sseld 2619	. . 3 |- (x e. On -> (A e. (R1` x) -> A e. (R1` suc x)))
9	8	reximia 2196	. 2 |- (E.x e. On A e. (R1` x) -> E.x e. On A e. (R1` suc x))
10	1, 9	syl 12	1 |- (A e. B -> E.x e. On A e. (R1` suc x))

Syntax hints:   -> wi 3   e. wcel 1300  E.wrex 2106   C_ wss 2593  Oncon0 3657  suc csuc 3659  ` cfv 3998  R1cr1 5748

This theorem is referenced by:  rankval 5779  rankon 5782  rankid 5783  rankr1 5785

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

Copyright terms: Public domain