Theorem tfi 7793

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 7803 or tfinds 7800 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 4082	. . . . . . . . 9 ⊢ (𝑥 ∈ (On ∖ 𝐴) → ¬ 𝑥 ∈ 𝐴)
2	1	adantl 481	. . . . . . . 8 ⊢ (((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ∧ 𝑥 ∈ (On ∖ 𝐴)) → ¬ 𝑥 ∈ 𝐴)
3		onss 7728	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
4		difin0ss 4323	. . . . . . . . . . . 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 4081	. . . . . . . . 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 3066	. . . . 5 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ∀𝑥 ∈ (On ∖ 𝐴) ¬ ((On ∖ 𝐴) ∩ 𝑥) = ∅)
13		ralnex 3060	. . . . 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 4316	. . . . . 6 ⊢ (On ⊆ 𝐴 ↔ (On ∖ 𝐴) = ∅)
16	15	necon3bbii 2977	. . . . 5 ⊢ (¬ On ⊆ 𝐴 ↔ (On ∖ 𝐴) ≠ ∅)
17		ordon 7720	. . . . . 6 ⊢ Ord On
18		difss 4086	. . . . . 6 ⊢ (On ∖ 𝐴) ⊆ On
19		tz7.5 6336	. . . . . 6 ⊢ ((Ord On ∧ (On ∖ 𝐴) ⊆ On ∧ (On ∖ 𝐴) ≠ ∅) → ∃𝑥 ∈ (On ∖ 𝐴)((On ∖ 𝐴) ∩ 𝑥) = ∅)
20	17, 18, 19	mp3an12 1453	. . . . 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 3947	. 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 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4283  Ord word 6314  Oncon0 6315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319

This theorem is referenced by:  tfisg  7794  tfis  7795  onsetrec  49895