Theorem wfii 13916

Description: The Principle of Well-Founded Induction. Theorem 6.27 of [TakeutiZaring] p. 32. This principle states that if B is a subclass of a well-ordered class A with the property that every element of B whose inital segment is included in A is itself equal to A.

Hypotheses

Ref	Expression
wfi.1	|- R We A
wfi.2	|- A.x e. A Pred(R, A, x) e. _V

Assertion

Ref	Expression
wfii	|- ((B C_ A /\ A.y e. A (Pred(R, A, y) C_ B -> y e. B)) -> A = B)

Distinct variable groups:   x,A   y,A   x,B   y,B   x,R   y,R

Proof of Theorem wfii

Step	Hyp	Ref	Expression
1		wfi.1	. . 3 |- R We A
2		wfi.2	. . 3 |- A.x e. A Pred(R, A, x) e. _V
3	1, 2	pm3.2i 307	. 2 |- (R We A /\ A.x e. A Pred(R, A, x) e. _V)
4		wfi 13915	. 2 |- (((R We A /\ A.x e. A Pred(R, A, x) e. _V) /\ (B C_ A /\ A.y e. A (Pred(R, A, y) C_ B -> y e. B))) -> A = B)
5	3, 4	mpan 759	1 |- ((B C_ A /\ A.y e. A (Pred(R, A, y) C_ B -> y e. B)) -> A = B)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593   We wwe 3624  Predcpred 13879

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-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-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-pred 13880

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