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Theorem tfi 4305
 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 2311 . . . . . . 7 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) ↔ ∀𝑥(𝑥 ∈ On → (𝑥𝐴𝑥𝐴)))
2 imdi 239 . . . . . . . 8 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ↔ ((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)))
32albii 1359 . . . . . . 7 (∀𝑥(𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ↔ ∀𝑥((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)))
41, 3bitri 173 . . . . . 6 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) ↔ ∀𝑥((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)))
5 dfss2 2934 . . . . . . . . . 10 (𝑥𝐴 ↔ ∀𝑦(𝑦𝑥𝑦𝐴))
65imbi2i 215 . . . . . . . . 9 ((𝑥 ∈ On → 𝑥𝐴) ↔ (𝑥 ∈ On → ∀𝑦(𝑦𝑥𝑦𝐴)))
7 19.21v 1753 . . . . . . . . 9 (∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) ↔ (𝑥 ∈ On → ∀𝑦(𝑦𝑥𝑦𝐴)))
86, 7bitr4i 176 . . . . . . . 8 ((𝑥 ∈ On → 𝑥𝐴) ↔ ∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
98imbi1i 227 . . . . . . 7 (((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) ↔ (∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
109albii 1359 . . . . . 6 (∀𝑥((𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) ↔ ∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
114, 10bitri 173 . . . . 5 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) ↔ ∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
12 simpl 102 . . . . . . . . . . 11 ((𝑦𝑥𝑥 ∈ On) → 𝑦𝑥)
13 tron 4119 . . . . . . . . . . . . . 14 Tr On
14 dftr2 3856 . . . . . . . . . . . . . 14 (Tr On ↔ ∀𝑦𝑥((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On))
1513, 14mpbi 133 . . . . . . . . . . . . 13 𝑦𝑥((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
1615spi 1429 . . . . . . . . . . . 12 𝑥((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
1716spi 1429 . . . . . . . . . . 11 ((𝑦𝑥𝑥 ∈ On) → 𝑦 ∈ On)
1812, 17jca 290 . . . . . . . . . 10 ((𝑦𝑥𝑥 ∈ On) → (𝑦𝑥𝑦 ∈ On))
1918imim1i 54 . . . . . . . . 9 (((𝑦𝑥𝑦 ∈ On) → 𝑦𝐴) → ((𝑦𝑥𝑥 ∈ On) → 𝑦𝐴))
20 impexp 250 . . . . . . . . 9 (((𝑦𝑥𝑦 ∈ On) → 𝑦𝐴) ↔ (𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)))
21 impexp 250 . . . . . . . . . 10 (((𝑦𝑥𝑥 ∈ On) → 𝑦𝐴) ↔ (𝑦𝑥 → (𝑥 ∈ On → 𝑦𝐴)))
22 bi2.04 237 . . . . . . . . . 10 ((𝑦𝑥 → (𝑥 ∈ On → 𝑦𝐴)) ↔ (𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
2321, 22bitri 173 . . . . . . . . 9 (((𝑦𝑥𝑥 ∈ On) → 𝑦𝐴) ↔ (𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
2419, 20, 233imtr3i 189 . . . . . . . 8 ((𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
2524alimi 1344 . . . . . . 7 (∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → ∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)))
2625imim1i 54 . . . . . 6 ((∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)) → (∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
2726alimi 1344 . . . . 5 (∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦𝑥𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)) → ∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
2811, 27sylbi 114 . . . 4 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
2928adantl 262 . . 3 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → ∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
30 sbim 1827 . . . . . . . . . 10 ([𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) ↔ ([𝑦 / 𝑥]𝑥 ∈ On → [𝑦 / 𝑥]𝑥𝐴))
31 clelsb3 2142 . . . . . . . . . . 11 ([𝑦 / 𝑥]𝑥 ∈ On ↔ 𝑦 ∈ On)
32 clelsb3 2142 . . . . . . . . . . 11 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
3331, 32imbi12i 228 . . . . . . . . . 10 (([𝑦 / 𝑥]𝑥 ∈ On → [𝑦 / 𝑥]𝑥𝐴) ↔ (𝑦 ∈ On → 𝑦𝐴))
3430, 33bitri 173 . . . . . . . . 9 ([𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) ↔ (𝑦 ∈ On → 𝑦𝐴))
3534ralbii 2330 . . . . . . . 8 (∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) ↔ ∀𝑦𝑥 (𝑦 ∈ On → 𝑦𝐴))
36 df-ral 2311 . . . . . . . 8 (∀𝑦𝑥 (𝑦 ∈ On → 𝑦𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)))
3735, 36bitri 173 . . . . . . 7 (∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) ↔ ∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)))
3837imbi1i 227 . . . . . 6 ((∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) ↔ (∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
3938albii 1359 . . . . 5 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) ↔ ∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)))
40 ax-setind 4262 . . . . 5 (∀𝑥(∀𝑦𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥𝐴) → (𝑥 ∈ On → 𝑥𝐴)) → ∀𝑥(𝑥 ∈ On → 𝑥𝐴))
4139, 40sylbir 125 . . . 4 (∀𝑥(∀𝑦(𝑦𝑥 → (𝑦 ∈ On → 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐴)) → ∀𝑥(𝑥 ∈ On → 𝑥𝐴))
42 dfss2 2934 . . . 4 (On ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ On → 𝑥𝐴))
4341, 42sylibr 137 . . 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 2960 . . 3 (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
4645biimpri 124 . 2 ((𝐴 ⊆ On ∧ On ⊆ 𝐴) → 𝐴 = On)
4744, 46syldan 266 1 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1241   = wceq 1243   ∈ wcel 1393  [wsb 1645  ∀wral 2306   ⊆ wss 2917  Tr wtr 3854  Oncon0 4100 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105 This theorem is referenced by:  tfis  4306
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