Theorem tfi 3089

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 included in 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:   x,A

Proof of Theorem tfi

Step	Hyp	Ref	Expression
1		eldifn 2134	. . . . . . . . 9 |- (x e. (On \ A) -> -. x e. A)
2	1	adantl 388	. . . . . . . 8 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> -. x e. A)
3		difin0ss 2303	. . . . . . . . . . . . 13 |- (((On \ A) i^i x) = (/) -> (x (_ On -> x (_ A))
4		onsst 2955	. . . . . . . . . . . . 13 |- (x e. On -> x (_ On)
5	3, 4	syl5com 52	. . . . . . . . . . . 12 |- (x e. On -> (((On \ A) i^i x) = (/) -> x (_ A))
6	5	imim1d 28	. . . . . . . . . . 11 |- (x e. On -> ((x (_ A -> x e. A) -> (((On \ A) i^i x) = (/) -> x e. A)))
7	6	a2i 9	. . . . . . . . . 10 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. On -> (((On \ A) i^i x) = (/) -> x e. A)))
8		eldifi 2133	. . . . . . . . . 10 |- (x e. (On \ A) -> x e. On)
9	7, 8	syl5 21	. . . . . . . . 9 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. (On \ A) -> (((On \ A) i^i x) = (/) -> x e. A)))
10	9	imp 350	. . . . . . . 8 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> (((On \ A) i^i x) = (/) -> x e. A))
11	2, 10	mtod 108	. . . . . . 7 |- (((x e. On -> (x (_ A -> x e. A)) /\ x e. (On \ A)) -> -. ((On \ A) i^i x) = (/))
12	11	ex 373	. . . . . 6 |- ((x e. On -> (x (_ A -> x e. A)) -> (x e. (On \ A) -> -. ((On \ A) i^i x) = (/)))
13	12	r19.20i2 1679	. . . . 5 |- (A.x e. On (x (_ A -> x e. A) -> A.x e. (On \ A) -. ((On \ A) i^i x) = (/))
14		ralnex 1629	. . . . 5 |- (A.x e. (On \ A) -. ((On \ A) i^i x) = (/) <-> -. E.x e. (On \ A)((On \ A) i^i x) = (/))
15	13, 14	sylib 198	. . . 4 |- (A.x e. On (x (_ A -> x e. A) -> -. E.x e. (On \ A)((On \ A) i^i x) = (/))
16		ssdif0 2298	. . . . . 6 |- (On (_ A <-> (On \ A) = (/))
17	16	necon3bbii 1573	. . . . 5 |- (-. On (_ A <-> (On \ A) =/= (/))
18		ordon 2950	. . . . . 6 |- Ord On
19		difss 2138	. . . . . 6 |- (On \ A) (_ On
20		tz7.5 2932	. . . . . 6 |- ((Ord On /\ (On \ A) (_ On /\ (On \ A) =/= (/)) -> E.x e. (On \ A)((On \ A) i^i x) = (/))
21	18, 19, 20	mp3an12 902	. . . . 5 |- ((On \ A) =/= (/) -> E.x e. (On \ A)((On \ A) i^i x) = (/))
22	17, 21	sylbi 199	. . . 4 |- (-. On (_ A -> E.x e. (On \ A)((On \ A) i^i x) = (/))
23	15, 22	nsyl2 118	. . 3 |- (A.x e. On (x (_ A -> x e. A) -> On (_ A)
24	23	anim2i 335	. 2 |- ((A (_ On /\ A.x e. On (x (_ A -> x e. A)) -> (A (_ On /\ On (_ A))
25		eqss 2048	. 2 |- (A = On <-> (A (_ On /\ On (_ A))
26	24, 25	sylibr 200	1 |- ((A (_ On /\ A.x e. On (x (_ A -> x e. A)) -> A = On)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 1099   e. wcel 1105   =/= wne 1561  A.wral 1621  E.wrex 1622   \ cdif 2015   i^i cin 2017   (_ wss 2018  (/)c0 2251  Ord word 2910  Oncon0 2911

This theorem is referenced by:  tfis 3090

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 773  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-tp 2386  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-tr 2649  df-eprel 2794  df-po 2804  df-so 2814  df-fr 2880  df-we 2897  df-ord 2914  df-on 2915

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