Theorem rankwflem 4665

Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 4664 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 4664	. 2 |- (A e. B -> E.x e. On A e. (R1` x))
2		suceloni 3062	. . . . 5 |- (x e. On -> suc x e. On)
3		visset 1813	. . . . . . 7 |- x e. V
4	3	sucid 3051	. . . . . 6 |- x e. suc x
5		r1ord2 4656	. . . . . 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 10	. . . 4 |- (x e. On -> (R1` x) (_ (R1` suc x))
8	7	sseld 2067	. . 3 |- (x e. On -> (A e. (R1` x) -> A e. (R1` suc x)))
9	8	r19.22i 1732	. 2 |- (E.x e. On A e. (R1` x) -> E.x e. On A e. (R1` suc x))
10	1, 9	syl 10	1 |- (A e. B -> E.x e. On A e. (R1` suc x))

Syntax hints:   -> wi 3   e. wcel 958  E.wrex 1646   (_ wss 2047  Oncon0 2948  suc csuc 2950  ` cfv 3182  R1cr1 4641

This theorem is referenced by:  rankval 4668  rankon 4671  rankid 4672  rankr1 4674

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

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