Theorem tfi 7548

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 7557 or tfinds 7554 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 4055	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
2	1	adantl 485	. . . . . . . 8 ⊢ (((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
3		onss 7485	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
4		difin0ss 4282	. . . . . . . . . . . 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 4054	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 𝐴) → 𝑥 ∈ On)
9	7, 8	impel 509	. . . . . . . 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 416	. . . . . 6 ⊢ ((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝑥 ∈ (On ∖ 𝐴) → ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅))
12	11	ralimi2 3125	. . . . 5 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
13		ralnex 3199	. . . . 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 4277	. . . . . 6 ⊢ (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
16	15	necon3bbii 3034	. . . . 5 ⊢ (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
17		ordon 7478	. . . . . 6 ⊢ Ord On
18		difss 4059	. . . . . 6 ⊢ (On ∖ 𝐴) ⊆ On
19		tz7.5 6180	. . . . . 6 ⊢ ((Ord On ∧ (On ∖ 𝐴) ⊆ On ∧ (On ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
20	17, 18, 19	mp3an12 1448	. . . . 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 143	. . 3 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → On ⊆ 𝐴)
23	22	anim2i 619	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → (𝐴 ⊆ On ∧ On ⊆ 𝐴))
24		eqss 3930	. 2 ⊢ (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
25	23, 24	sylibr 237	1 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  Ord word 6158  Oncon0 6159

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163

This theorem is referenced by:  tfis  7549  tfisg  33168  onsetrec  45237