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Theorem tfi 4503
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 (𝑥𝐴𝑥𝐴)) → 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 2422 . . . . . . 7 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) ↔ ∀𝑥(𝑥 ∈ On → (𝑥𝐴𝑥𝐴)))
2 imdi 249 . . . . . . . 8 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ↔ ((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)))
32albii 1447 . . . . . . 7 (∀𝑥(𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ↔ ∀𝑥((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)))
41, 3bitri 183 . . . . . 6 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) ↔ ∀𝑥((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)))
5 dfss2 3090 . . . . . . . . . 10 (𝑥𝐴 ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
65imbi2i 225 . . . . . . . . 9 ((𝑥 ∈ On → 𝑥𝐴) ↔ (𝑥 ∈ On → ∀𝑦(𝑦𝑥𝑦𝐴)))
7 19.21v 1846 . . . . . . . . 9 (∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) ↔ (𝑥 ∈ On → ∀𝑦(𝑦𝑥𝑦𝐴)))
86, 7bitr4i 186 . . . . . . . 8 ((𝑥 ∈ On → 𝑥𝐴) ↔ ∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
98imbi1i 237 . . . . . . 7 (((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) ↔ (∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
109albii 1447 . . . . . 6 (∀𝑥((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) ↔ ∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
114, 10bitri 183 . . . . 5 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) ↔ ∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
12 simpl 108 . . . . . . . . . . 11 ((𝑦𝑥𝑥 ∈ On) → 𝑦𝑥)
13 tron 4311 . . . . . . . . . . . . . 14 Tr On
14 dftr2 4035 . . . . . . . . . . . . . 14 (Tr On ↔ ∀𝑦𝑥((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On))
1513, 14mpbi 144 . . . . . . . . . . . . 13 𝑦𝑥((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
1615spi 1517 . . . . . . . . . . . 12 𝑥((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
1716spi 1517 . . . . . . . . . . 11 ((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
1812, 17jca 304 . . . . . . . . . 10 ((𝑦𝑥𝑥 ∈ On) → (𝑦𝑥𝑦 ∈ On))
1918imim1i 60 . . . . . . . . 9 (((𝑦𝑥𝑦 ∈ On) → 𝑦𝐴) → ((𝑦𝑥𝑥 ∈ On) → 𝑦𝐴))
20 impexp 261 . . . . . . . . 9 (((𝑦𝑥𝑦 ∈ On) → 𝑦𝐴) ↔ (𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)))
21 impexp 261 . . . . . . . . . 10 (((𝑦𝑥𝑥 ∈ On) → 𝑦𝐴) ↔ (𝑦𝑥 → (𝑥 ∈ On → 𝑦𝐴)))
22 bi2.04 247 . . . . . . . . . 10 ((𝑦𝑥 → (𝑥 ∈ On → 𝑦𝐴)) ↔ (𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
2321, 22bitri 183 . . . . . . . . 9 (((𝑦𝑥𝑥 ∈ On) → 𝑦𝐴) ↔ (𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
2419, 20, 233imtr3i 199 . . . . . . . 8 ((𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
2524alimi 1432 . . . . . . 7 (∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → ∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
2625imim1i 60 . . . . . 6 ((∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)) → (∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
2726alimi 1432 . . . . 5 (∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)) → ∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
2811, 27sylbi 120 . . . 4 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
2928adantl 275 . . 3 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → ∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
30 sbim 1927 . . . . . . . . . 10 ([𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) ↔ ([𝑦 / 𝑥]𝑥 ∈ On → [𝑦 / 𝑥]𝑥𝐴))
31 clelsb3 2245 . . . . . . . . . . 11 ([𝑦 / 𝑥]𝑥 ∈ On ↔ 𝑦 ∈ On)
32 clelsb3 2245 . . . . . . . . . . 11 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
3331, 32imbi12i 238 . . . . . . . . . 10 (([𝑦 / 𝑥]𝑥 ∈ On → [𝑦 / 𝑥]𝑥𝐴) ↔ (𝑦 ∈ On → 𝑦𝐴))
3430, 33bitri 183 . . . . . . . . 9 ([𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) ↔ (𝑦 ∈ On → 𝑦𝐴))
3534ralbii 2444 . . . . . . . 8 (∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) ↔ ∀𝑦𝑥 (𝑦 ∈ On → 𝑦𝐴))
36 df-ral 2422 . . . . . . . 8 (∀𝑦𝑥 (𝑦 ∈ On → 𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)))
3735, 36bitri 183 . . . . . . 7 (∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)))
3837imbi1i 237 . . . . . 6 ((∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) ↔ (∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
3938albii 1447 . . . . 5 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) ↔ ∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
40 ax-setind 4459 . . . . 5 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) → ∀𝑥(𝑥 ∈ On → 𝑥𝐴))
4139, 40sylbir 134 . . . 4 (∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)) → ∀𝑥(𝑥 ∈ On → 𝑥𝐴))
42 dfss2 3090 . . . 4 (On ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ On → 𝑥𝐴))
4341, 42sylibr 133 . . 3 (∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)) → On ⊆ 𝐴)
4429, 43syl 14 . 2 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → On ⊆ 𝐴)
45 eqss 3116 . . 3 (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
4645biimpri 132 . 2 ((𝐴 ⊆ On ∧ On ⊆ 𝐴) → 𝐴 = On)
4744, 46syldan 280 1 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1330   = wceq 1332  wcel 1481  [wsb 1736  wral 2417  wss 3075  Tr wtr 4033  Oncon0 4292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3081  df-ss 3088  df-uni 3744  df-tr 4034  df-iord 4295  df-on 4297
This theorem is referenced by:  tfis  4504
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