Theorem tfi 7875

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 7885 or tfinds 7882 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 4131	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
2	1	adantl 481	. . . . . . . 8 ⊢ (((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
3		onss 7806	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
4		difin0ss 4372	. . . . . . . . . . . 12 ⊢ (((On ∖ 𝐴) ∩ 𝑥) = ∅ → (𝑥 ⊆ On → 𝑥 ⊆ 𝐴))
5	3, 4	syl5com 31	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥 ⊆ 𝐴))
6	5	imim1d 82	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ((𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥 ∈ 𝐴)))
7	6	a2i 14	. . . . . . . . 9 ⊢ ((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ On → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥 ∈ 𝐴)))
8		eldifi 4130	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 𝐴) → 𝑥 ∈ On)
9	7, 8	impel 505	. . . . . . . 8 ⊢ (((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → (((On ∖ 𝐴) ∩ 𝑥) = ∅ → 𝑥 ∈ 𝐴))
10	2, 9	mtod 198	. . . . . . 7 ⊢ (((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
11	10	ex 412	. . . . . 6 ⊢ ((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅))
12	11	ralimi2 3077	. . . . 5 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
13		ralnex 3071	. . . . 5 ⊢ (∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅ ↔ ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
14	12, 13	sylib 218	. . . 4 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ¬ ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
15		ssdif0 4365	. . . . . 6 ⊢ (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
16	15	necon3bbii 2987	. . . . 5 ⊢ (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
17		ordon 7798	. . . . . 6 ⊢ Ord On
18		difss 4135	. . . . . 6 ⊢ (On ∖ 𝐴) ⊆ On
19		tz7.5 6404	. . . . . 6 ⊢ ((Ord On ∧ (On ∖ 𝐴) ⊆ On ∧ (On ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
20	17, 18, 19	mp3an12 1452	. . . . 5 ⊢ ((On ∖ 𝐴) ≠ ∅ → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
21	16, 20	sylbi 217	. . . 4 ⊢ (¬ On ⊆ 𝐴 → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
22	14, 21	nsyl2 141	. . 3 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → On ⊆ 𝐴)
23	22	anim2i 617	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝐴 ⊆ On ∧ On ⊆ 𝐴))
24		eqss 3998	. 2 ⊢ (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
25	23, 24	sylibr 234	1 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ≠ wne 2939  ∀wral 3060  ∃wrex 3069   ∖ cdif 3947   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  Ord word 6382  Oncon0 6383

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387

This theorem is referenced by:  tfisg  7876  tfis  7877  onsetrec  49282