Theorem rankwflem 4637

Description: Every set is well-founded, assuming the Axiom of Regularity. Proposition 9.13 of [TakeutiZaring] p. 78. This variant of tz9.13g 4636 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 4636	. 2 |- (A e. B -> E.x e. On A e. (R1` x))
2		suceloni 3052	. . . . 5 |- (x e. On -> suc x e. On)
3		visset 1804	. . . . . . 7 |- x e. V
4	3	sucid 3041	. . . . . 6 |- x e. suc x
5		r1ord2 4628	. . . . . 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 2057	. . 3 |- (x e. On -> (A e. (R1` x) -> A e. (R1` suc x)))
9	8	r19.22i 1724	. 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 955  E.wrex 1638   (_ wss 2037  Oncon0 2938  suc csuc 2940  ` cfv 3172  R1cr1 4613

This theorem is referenced by:  rankval 4640  rankon 4643  rankid 4644  rankr1 4646

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-reg 4565  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188  df-rdg 3917  df-r1 4615

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