Theorem tz9.13g 4664

Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13 4663 expresses the class existence requirement as an antecedent.

Assertion

Ref	Expression
tz9.13g	|- (A e. B -> E.x e. On A e. (R1` x))

Distinct variable group:   x,A

Proof of Theorem tz9.13g

Step	Hyp	Ref	Expression
1		ax-17 971	. . 3 |- (y = A -> A.x y = A)
2		eleq1 1534	. . 3 |- (y = A -> (y e. (R1` x) <-> A e. (R1` x)))
3	1, 2	rexbid 1662	. 2 |- (y = A -> (E.x e. On y e. (R1` x) <-> E.x e. On A e. (R1` x)))
4		visset 1813	. . 3 |- y e. V
5	4	tz9.13 4663	. 2 |- E.x e. On y e. (R1` x)
6	3, 5	vtoclg 1847	1 |- (A e. B -> E.x e. On A e. (R1` x))

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  E.wrex 1646  Oncon0 2948  ` cfv 3182  R1cr1 4641

This theorem is referenced by:  rankwflem 4665

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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