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Theorem tfi 7863
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 7873 or tfinds 7870 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 4127 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥𝐴)
21adantl 480 . . . . . . . 8 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥𝐴)
3 onss 7793 . . . . . . . . . . . 12 (𝑥 ∈ On → 𝑥 ⊆ On)
4 difin0ss 4373 . . . . . . . . . . . 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 4126 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → 𝑥 ∈ On)
97, 8impel 504 . . . . . . . 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 411 . . . . . 6 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅))
1211ralimi2 3068 . . . . 5 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
13 ralnex 3062 . . . . 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 4366 . . . . . 6 (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
1615necon3bbii 2978 . . . . 5 (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
17 ordon 7785 . . . . . 6 Ord On
18 difss 4131 . . . . . 6 (On ∖ 𝐴) ⊆ On
19 tz7.5 6397 . . . . . 6 ((Ord On ∧ (On ∖ 𝐴) ⊆ On ∧ (On ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2017, 18, 19mp3an12 1448 . . . . 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 615 . 2 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → (𝐴 ⊆ On ∧ On ⊆ 𝐴))
24 eqss 3995 . 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 394   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  cdif 3944  cin 3946  wss 3947  c0 4325  Ord word 6375  Oncon0 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-tr 5271  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6379  df-on 6380
This theorem is referenced by:  tfisg  7864  tfis  7865  onsetrec  48454
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