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Theorem tfi 4228
 Description: The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinal numbers with the property that every ordinal number included in A also belongs to A, then every ordinal number is in A. (Contributed by NM, 18-Feb-2004.)
Assertion
Ref Expression
tfi ((A ⊆ On x On (xAx A)) → A = On)
Distinct variable group:   x,A

Proof of Theorem tfi
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-ral 2285 . . . . . . 7 (x On (xAx A) ↔ x(x On → (xAx A)))
2 imdi 239 . . . . . . . 8 ((x On → (xAx A)) ↔ ((x On → xA) → (x On → x A)))
32albii 1335 . . . . . . 7 (x(x On → (xAx A)) ↔ x((x On → xA) → (x On → x A)))
41, 3bitri 173 . . . . . 6 (x On (xAx A) ↔ x((x On → xA) → (x On → x A)))
5 dfss2 2907 . . . . . . . . . 10 (xAy(y xy A))
65imbi2i 215 . . . . . . . . 9 ((x On → xA) ↔ (x On → y(y xy A)))
7 19.21v 1731 . . . . . . . . 9 (y(x On → (y xy A)) ↔ (x On → y(y xy A)))
86, 7bitr4i 176 . . . . . . . 8 ((x On → xA) ↔ y(x On → (y xy A)))
98imbi1i 227 . . . . . . 7 (((x On → xA) → (x On → x A)) ↔ (y(x On → (y xy A)) → (x On → x A)))
109albii 1335 . . . . . 6 (x((x On → xA) → (x On → x A)) ↔ x(y(x On → (y xy A)) → (x On → x A)))
114, 10bitri 173 . . . . 5 (x On (xAx A) ↔ x(y(x On → (y xy A)) → (x On → x A)))
12 ax-ia1 99 . . . . . . . . . . 11 ((y x x On) → y x)
13 tron 4064 . . . . . . . . . . . . . 14 Tr On
14 dftr2 3826 . . . . . . . . . . . . . 14 (Tr On ↔ yx((y x x On) → y On))
1513, 14mpbi 133 . . . . . . . . . . . . 13 yx((y x x On) → y On)
1615spi 1407 . . . . . . . . . . . 12 x((y x x On) → y On)
1716spi 1407 . . . . . . . . . . 11 ((y x x On) → y On)
1812, 17jca 290 . . . . . . . . . 10 ((y x x On) → (y x y On))
1918imim1i 54 . . . . . . . . 9 (((y x y On) → y A) → ((y x x On) → y A))
20 impexp 250 . . . . . . . . 9 (((y x y On) → y A) ↔ (y x → (y On → y A)))
21 impexp 250 . . . . . . . . . 10 (((y x x On) → y A) ↔ (y x → (x On → y A)))
22 bi2.04 237 . . . . . . . . . 10 ((y x → (x On → y A)) ↔ (x On → (y xy A)))
2321, 22bitri 173 . . . . . . . . 9 (((y x x On) → y A) ↔ (x On → (y xy A)))
2419, 20, 233imtr3i 189 . . . . . . . 8 ((y x → (y On → y A)) → (x On → (y xy A)))
2524alimi 1320 . . . . . . 7 (y(y x → (y On → y A)) → y(x On → (y xy A)))
2625imim1i 54 . . . . . 6 ((y(x On → (y xy A)) → (x On → x A)) → (y(y x → (y On → y A)) → (x On → x A)))
2726alimi 1320 . . . . 5 (x(y(x On → (y xy A)) → (x On → x A)) → x(y(y x → (y On → y A)) → (x On → x A)))
2811, 27sylbi 114 . . . 4 (x On (xAx A) → x(y(y x → (y On → y A)) → (x On → x A)))
2928adantl 262 . . 3 ((A ⊆ On x On (xAx A)) → x(y(y x → (y On → y A)) → (x On → x A)))
30 sbim 1805 . . . . . . . . . 10 ([y / x](x On → x A) ↔ ([y / x]x On → [y / x]x A))
31 clelsb3 2120 . . . . . . . . . . 11 ([y / x]x On ↔ y On)
32 clelsb3 2120 . . . . . . . . . . 11 ([y / x]x Ay A)
3331, 32imbi12i 228 . . . . . . . . . 10 (([y / x]x On → [y / x]x A) ↔ (y On → y A))
3430, 33bitri 173 . . . . . . . . 9 ([y / x](x On → x A) ↔ (y On → y A))
3534ralbii 2304 . . . . . . . 8 (y x [y / x](x On → x A) ↔ y x (y On → y A))
36 df-ral 2285 . . . . . . . 8 (y x (y On → y A) ↔ y(y x → (y On → y A)))
3735, 36bitri 173 . . . . . . 7 (y x [y / x](x On → x A) ↔ y(y x → (y On → y A)))
3837imbi1i 227 . . . . . 6 ((y x [y / x](x On → x A) → (x On → x A)) ↔ (y(y x → (y On → y A)) → (x On → x A)))
3938albii 1335 . . . . 5 (x(y x [y / x](x On → x A) → (x On → x A)) ↔ x(y(y x → (y On → y A)) → (x On → x A)))
40 ax-setind 4200 . . . . 5 (x(y x [y / x](x On → x A) → (x On → x A)) → x(x On → x A))
4139, 40sylbir 125 . . . 4 (x(y(y x → (y On → y A)) → (x On → x A)) → x(x On → x A))
42 dfss2 2907 . . . 4 (On ⊆ Ax(x On → x A))
4341, 42sylibr 137 . . 3 (x(y(y x → (y On → y A)) → (x On → x A)) → On ⊆ A)
4429, 43syl 14 . 2 ((A ⊆ On x On (xAx A)) → On ⊆ A)
45 eqss 2933 . . 3 (A = On ↔ (A ⊆ On On ⊆ A))
4645biimpri 124 . 2 ((A ⊆ On On ⊆ A) → A = On)
4744, 46syldan 266 1 ((A ⊆ On x On (xAx A)) → A = On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1224   = wceq 1226   ∈ wcel 1370  [wsb 1623  ∀wral 2280   ⊆ wss 2890  Tr wtr 3824  Oncon0 4045 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-setind 4200 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-in 2897  df-ss 2904  df-uni 3551  df-tr 3825  df-iord 4048  df-on 4050 This theorem is referenced by:  tfis  4229
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