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Theorem tfi 7200
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 7209 or tfinds 7206 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 3884 . . . . . . . . 9 (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥𝐴)
21adantl 467 . . . . . . . 8 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥𝐴)
3 eldifi 3883 . . . . . . . . . 10 (𝑥 ∈ (On ∖ 𝐴) → 𝑥 ∈ On)
4 onss 7137 . . . . . . . . . . . . 13 (𝑥 ∈ On → 𝑥 ⊆ On)
5 difin0ss 4093 . . . . . . . . . . . . 13 (((On ∖ 𝐴) ∩ 𝑥) = ∅ → (𝑥 ⊆ On → 𝑥𝐴))
64, 5syl5com 31 . . . . . . . . . . . 12 (𝑥 ∈ On → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴))
76imim1d 82 . . . . . . . . . . 11 (𝑥 ∈ On → ((𝑥𝐴𝑥𝐴) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴)))
87a2i 14 . . . . . . . . . 10 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) → (𝑥 ∈ On → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴)))
93, 8syl5 34 . . . . . . . . 9 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴)))
109imp 393 . . . . . . . 8 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥𝐴))
112, 10mtod 189 . . . . . . 7 (((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
1211ex 397 . . . . . 6 ((𝑥 ∈ On → (𝑥𝐴𝑥𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅))
1312ralimi2 3098 . . . . 5 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
14 ralnex 3141 . . . . 5 (∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅ ↔ ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
1513, 14sylib 208 . . . 4 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
16 ssdif0 4089 . . . . . 6 (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
1716necon3bbii 2990 . . . . 5 (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
18 ordon 7129 . . . . . 6 Ord On
19 difss 3888 . . . . . 6 (On ∖ 𝐴) ⊆ On
20 tz7.5 5887 . . . . . 6 ((Ord On ∧ (On ∖ 𝐴) ⊆ On ∧ (On ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2118, 19, 20mp3an12 1562 . . . . 5 ((On ∖ 𝐴) ≠ ∅ → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2217, 21sylbi 207 . . . 4 (¬ On ⊆ 𝐴 → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
2315, 22nsyl2 144 . . 3 (∀𝑥 ∈ On (𝑥𝐴𝑥𝐴) → On ⊆ 𝐴)
2423anim2i 603 . 2 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → (𝐴 ⊆ On ∧ On ⊆ 𝐴))
25 eqss 3767 . 2 (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
2624, 25sylibr 224 1 ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥𝐴𝑥𝐴)) → 𝐴 = On)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wral 3061  wrex 3062  cdif 3720  cin 3722  wss 3723  c0 4063  Ord word 5865  Oncon0 5866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-ord 5869  df-on 5870
This theorem is referenced by:  tfis  7201  tfisg  32052  onsetrec  42982
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