Theorem rankwflem 3509

Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 3508 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(s):   x,A

Proof of Theorem rankwflem

Step	Hyp	Ref	Expression
1		tz9.13g 3508	. 2 |- (A e. B -> E.x e. On A e. (R1` x))
2		suceloni 2314	. . . . 5 |- (x e. On -> suc x e. On)
3		visset 1350	. . . . . . 7 |- x e. V
4	3	sucid 2304	. . . . . 6 |- x e. suc x
5		r1ord2 3500	. . . . . 6 |- (suc x e. On -> (x e. suc x -> (R1` x) (_ (R1` suc x)))
6	4, 5	mpi 44	. . . . 5 |- (suc x e. On -> (R1` x) (_ (R1` suc x))
7	2, 6	syl 12	. . . 4 |- (x e. On -> (R1` x) (_ (R1` suc x))
8	7	sseld 1506	. . 3 |- (x e. On -> (A e. (R1` x) -> A e. (R1` suc x)))
9	8	r19.22i 1273	. 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 2   e. wcel 1092  E.wrex 1202   (_ wss 1487  Oncon0 2199  suc csuc 2201  ` cfv 2422  R1cr1 3485

This theorem is referenced by:  rankval 3512  rankon 3515  rankid 3516  rankr1 3518

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487

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