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Theorem tfi 7838
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 7848 or tfinds 7845 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 4126 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥𝐴)
21adantl 482 . . . . . . . 8 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥𝐴)
3 onss 7768 . . . . . . . . . . . 12 (𝑥 ∈ On → 𝑥 ⊆ On)
4 difin0ss 4367 . . . . . . . . . . . 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 4125 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → 𝑥 ∈ On)
97, 8impel 506 . . . . . . . 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 413 . . . . . 6 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅))
1211ralimi2 3078 . . . . 5 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
13 ralnex 3072 . . . . 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 4362 . . . . . 6 (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
1615necon3bbii 2988 . . . . 5 (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
17 ordon 7760 . . . . . 6 Ord On
18 difss 4130 . . . . . 6 (On ∖ 𝐴) ⊆ On
19 tz7.5 6382 . . . . . 6 ((Ord On ∧ (On ∖ 𝐴) ⊆ On ∧ (On ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2017, 18, 19mp3an12 1451 . . . . 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 617 . 2 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → (𝐴 ⊆ On ∧ On ⊆ 𝐴))
24 eqss 3996 . 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 396   = wceq 1541  wcel 2106  wne 2940  wral 3061  wrex 3070  cdif 3944  cin 3946  wss 3947  c0 4321  Ord word 6360  Oncon0 6361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365
This theorem is referenced by:  tfisg  7839  tfis  7840  onsetrec  47706
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