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Theorem tfi 7562
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 7571 or tfinds 7568 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 4090 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥𝐴)
21adantl 485 . . . . . . . 8 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥𝐴)
3 onss 7499 . . . . . . . . . . . 12 (𝑥 ∈ On → 𝑥 ⊆ On)
4 difin0ss 4311 . . . . . . . . . . . 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 4089 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → 𝑥 ∈ On)
97, 8impel 509 . . . . . . . 8 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴))
102, 9mtod 201 . . . . . . 7 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
1110ex 416 . . . . . 6 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅))
1211ralimi2 3152 . . . . 5 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
13 ralnex 3230 . . . . 5 (∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅ ↔ ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
1412, 13sylib 221 . . . 4 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
15 ssdif0 4306 . . . . . 6 (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
1615necon3bbii 3061 . . . . 5 (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
17 ordon 7492 . . . . . 6 Ord On
18 difss 4094 . . . . . 6 (On ∖ 𝐴) ⊆ On
19 tz7.5 6199 . . . . . 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 220 . . . 4 (¬ On ⊆ 𝐴 → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2214, 21nsyl2 143 . . 3 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → On ⊆ 𝐴)
2322anim2i 619 . 2 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → (𝐴 ⊆ On ∧ On ⊆ 𝐴))
24 eqss 3968 . 2 (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
2523, 24sylibr 237 1 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  cdif 3916  cin 3918  wss 3919  c0 4276  Ord word 6177  Oncon0 6178
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182
This theorem is referenced by:  tfis  7563  tfisg  33112  onsetrec  45163
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