Theorem tfi 7898

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 7908 or tfinds 7905 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

Assertion

Ref	Expression
tfi	⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)

Distinct variable group:   𝑥,𝐴

Proof of Theorem tfi

Step	Hyp	Ref	Expression
1		eldifn 4071	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
2	1	adantl 487	. . . . . . . 8 ⊢ (((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
3		onss 7833	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
4		difin0ss 4324	. . . . . . . . . . . 12 ⊢ (((On ∖ 𝐴) ∩ 𝑥) = ∅ → (𝑥 ⊆ On → 𝑥 ⊆ 𝐴))
5	3, 4	syl5com 32	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐴))
6	5	imim1d 83	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ((𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥 ∈ 𝐴)))
7	6	a2i 15	. . . . . . . . 9 ⊢ ((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ On → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥 ∈ 𝐴)))
8		eldifi 4070	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 𝐴) → 𝑥 ∈ On)
9	7, 8	impel 515	. . . . . . . 8 ⊢ (((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥 ∈ 𝐴))
10	2, 9	mtod 201	. . . . . . 7 ⊢ (((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
11	10	ex 418	. . . . . 6 ⊢ ((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅))
12	11	ralimi2 3098	. . . . 5 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
13		ralnex 3092	. . . . 5 ⊢ (∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅ ↔ ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
14	12, 13	sylib 221	. . . 4 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
15		ssdif0 4317	. . . . . 6 ⊢ (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
16	15	necon3bbii 3006	. . . . 5 ⊢ (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
17		ordon 7823	. . . . . 6 ⊢ Ord On
18		difss 4075	. . . . . 6 ⊢ (On ∖ 𝐴) ⊆ On
19		tz7.5 6409	. . . . . 6 ⊢ ((Ord On ∧ (On ∖ 𝐴) ⊆ On ∧ (On ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
20	17, 18, 19	mp3an12 1481	. . . . 5 ⊢ ((On ∖ 𝐴) ≠ ∅ → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
21	16, 20	sylbi 220	. . . 4 ⊢ (¬ On ⊆ 𝐴 → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
22	14, 21	nsyl2 142	. . 3 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → On ⊆ 𝐴)
23	22	anim2i 629	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝐴 ⊆ On ∧ On ⊆ 𝐴))
24		eqss 3928	. 2 ⊢ (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
25	23, 24	sylibr 237	1 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  ∅c0 4281  Ord word 6387  Oncon0 6388

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392

This theorem is used by:  tfisg  7899  tfis  7900  onsetrec  51207