Theorem tfi 2244

Description: The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinals with the property that every ordinal number that is a subset of A also belongs to A, then every ordinal is in A.

Assertion

Ref	Expression
tfi	|- ((A (_ On /\ A.x e. On (x (_ A -> x e. A)) -> A = On)

Distinct variable group(s):   x,A

Proof of Theorem tfi

Step	Hyp	Ref	Expression
1		eldifn 1592	. . . . . . . . . 10 |- (x e. (On \ A) -> -. x e. A)
2	1	adantl 305	. . . . . . . . 9 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> -. x e. A)
3		onsst 2243	. . . . . . . . . . . . . 14 |- (x e. On -> x (_ On)
4		difin0ss 1753	. . . . . . . . . . . . . . 15 |- (((On \ A) i^i x) = (/) -> (x (_ On -> x (_ A))
5	4	com12 13	. . . . . . . . . . . . . 14 |- (x (_ On -> (((On \ A) i^i x) = (/) -> x (_ A))
6	3, 5	syl 12	. . . . . . . . . . . . 13 |- (x e. On -> (((On \ A) i^i x) = (/) -> x (_ A))
7	6	syl4d 28	. . . . . . . . . . . 12 |- (x e. On -> ((x (_ A -> x e. A) -> (((On \ A) i^i x) = (/) -> x e. A)))
8	7	a2i 8	. . . . . . . . . . 11 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. On -> (((On \ A) i^i x) = (/) -> x e. A)))
9		eldifi 1591	. . . . . . . . . . 11 |- (x e. (On \ A) -> x e. On)
10	8, 9	syl5 22	. . . . . . . . . 10 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. (On \ A) -> (((On \ A) i^i x) = (/) -> x e. A)))
11	10	imp 277	. . . . . . . . 9 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> (((On \ A) i^i x) = (/) -> x e. A))
12	2, 11	mtod 95	. . . . . . . 8 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> -. ((On \ A) i^i x) = (/))
13	12	exp 291	. . . . . . 7 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. (On \ A) -> -. ((On \ A) i^i x) = (/)))
14	13	19.20i 691	. . . . . 6 |- (A.x(x e. On -> (x (_ A -> x e. A)) -> A.x(x e. (On \ A) -> -. ((On \ A) i^i x) = (/)))
15		df-ral 1205	. . . . . 6 |- (A.x e. On (x (_ A -> x e. A) <-> A.x(x e. On -> (x (_ A -> x e. A)))
16		df-ral 1205	. . . . . 6 |- (A.x e. (On \ A) -. ((On \ A) i^i x) = (/) <-> A.x(x e. (On \ A) -> -. ((On \ A) i^i x) = (/)))
17	14, 15, 16	3imtr4 192	. . . . 5 |- (A.x e. On (x (_ A -> x e. A) -> A.x e. (On \ A) -. ((On \ A) i^i x) = (/))
18		ralnex 1209	. . . . 5 |- (A.x e. (On \ A) -. ((On \ A) i^i x) = (/) <-> -. E.x e. (On \ A)((On \ A) i^i x) = (/))
19	17, 18	sylib 173	. . . 4 |- (A.x e. On (x (_ A -> x e. A) -> -. E.x e. (On \ A)((On \ A) i^i x) = (/))
20		ssdif0 1748	. . . . . 6 |- (On (_ A <-> (On \ A) = (/))
21	20	negbii 162	. . . . 5 |- (-. On (_ A <-> -. (On \ A) = (/))
22		difss 1596	. . . . . 6 |- (On \ A) (_ On
23		ordon 2238	. . . . . . 7 |- Ord On
24		tz7.5 2220	. . . . . . 7 |- ((Ord On /\ ((On \ A) (_ On /\ -. (On \ A) = (/))) -> E.x e. (On \ A)((On \ A) i^i x) = (/))
25	23, 24	mpan 518	. . . . . 6 |- (((On \ A) (_ On /\ -. (On \ A) = (/)) -> E.x e. (On \ A)((On \ A) i^i x) = (/))
26	22, 25	mpan 518	. . . . 5 |- (-. (On \ A) = (/) -> E.x e. (On \ A)((On \ A) i^i x) = (/))
27	21, 26	sylbi 174	. . . 4 |- (-. On (_ A -> E.x e. (On \ A)((On \ A) i^i x) = (/))
28	19, 27	nsyl2 103	. . 3 |- (A.x e. On (x (_ A -> x e. A) -> On (_ A)
29	28	anim2i 270	. 2 |- ((A (_ On /\ A.x e. On (x (_ A -> x e. A)) -> (A (_ On /\ On (_ A))
30		eqss 1516	. 2 |- (A = On <-> (A (_ On /\ On (_ A))
31	29, 30	sylibr 175	1 |- ((A (_ On /\ A.x e. On (x (_ A -> x e. A)) -> A = On)

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202   \ cdif 1484   i^i cin 1486   (_ wss 1487  (/)c0 1707  Ord word 2198  Oncon0 2199

This theorem is referenced by:  tfis 2245

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

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-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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