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Theorem tfi 7794
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.

See Theorem tfindes 7804 or tfinds 7801 for the version involving basis and induction hypotheses. (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
StepHypRef Expression
1 eldifn 4092 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥𝐴)
21adantl 483 . . . . . . . 8 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥𝐴)
3 onss 7724 . . . . . . . . . . . 12 (𝑥 ∈ On → 𝑥 ⊆ On)
4 difin0ss 4333 . . . . . . . . . . . 12 (((On ∖ 𝐴) ∩ 𝑥) = ∅ → (𝑥 ⊆ On → 𝑥𝐴))
53, 4syl5com 31 . . . . . . . . . . 11 (𝑥 ∈ On → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴))
65imim1d 82 . . . . . . . . . 10 (𝑥 ∈ On → ((𝑥𝐴𝑥𝐴) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴)))
76a2i 14 . . . . . . . . 9 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) → (𝑥 ∈ On → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴)))
8 eldifi 4091 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → 𝑥 ∈ On)
97, 8impel 507 . . . . . . . 8 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴))
102, 9mtod 197 . . . . . . 7 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
1110ex 414 . . . . . 6 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅))
1211ralimi2 3082 . . . . 5 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
13 ralnex 3076 . . . . 5 (∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅ ↔ ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
1412, 13sylib 217 . . . 4 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
15 ssdif0 4328 . . . . . 6 (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
1615necon3bbii 2992 . . . . 5 (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
17 ordon 7716 . . . . . 6 Ord On
18 difss 4096 . . . . . 6 (On ∖ 𝐴) ⊆ On
19 tz7.5 6343 . . . . . 6 ((Ord On ∧ (On ∖ 𝐴) ⊆ On ∧ (On ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2017, 18, 19mp3an12 1452 . . . . 5 ((On ∖ 𝐴) ≠ ∅ → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2116, 20sylbi 216 . . . 4 (¬ On ⊆ 𝐴 → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2214, 21nsyl2 141 . . 3 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → On ⊆ 𝐴)
2322anim2i 618 . 2 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → (𝐴 ⊆ On ∧ On ⊆ 𝐴))
24 eqss 3964 . 2 (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
2523, 24sylibr 233 1 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1542  wcel 2107  wne 2944  wral 3065  wrex 3074  cdif 3912  cin 3914  wss 3915  c0 4287  Ord word 6321  Oncon0 6322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-ord 6325  df-on 6326
This theorem is referenced by:  tfisg  7795  tfis  7796  onsetrec  47227
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