Theorem tfi 2244

Description: The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinals with the property that every ordinal number that is a subset of A also belongs to A, then every ordinal is in A.

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem tfi

Step	Hyp	Ref	Expression
1		eldifn 1592	. . . . . . . . . 10 ⊢ (x ∈ (On ∖ A) → ¬ x ∈ A)
2	1	adantl 305	. . . . . . . . 9 ⊢ (((x ∈ On → (x ⊆ A → x ∈ A)) ∧ x ∈ (On ∖ A)) → ¬ x ∈ A)
3		onsst 2243	. . . . . . . . . . . . . 14 ⊢ (x ∈ On → x ⊆ On)
4		difin0ss 1753	. . . . . . . . . . . . . . 15 ⊢ (((On ∖ A) ∩ x) = ∅ → (x ⊆ On → x ⊆ A))
5	4	com12 13	. . . . . . . . . . . . . 14 ⊢ (x ⊆ On → (((On ∖ A) ∩ x) = ∅ → x ⊆ A))
6	3, 5	syl 12	. . . . . . . . . . . . 13 ⊢ (x ∈ On → (((On ∖ A) ∩ x) = ∅ → x ⊆ A))
7	6	syl4d 28	. . . . . . . . . . . 12 ⊢ (x ∈ On → ((x ⊆ A → x ∈ A) → (((On ∖ A) ∩ x) = ∅ → x ∈ A)))
8	7	a2i 8	. . . . . . . . . . 11 ⊢ ((x ∈ On → (x ⊆ A → x ∈ A)) → (x ∈ On → (((On ∖ A) ∩ x) = ∅ → x ∈ A)))
9		eldifi 1591	. . . . . . . . . . 11 ⊢ (x ∈ (On ∖ A) → x ∈ On)
10	8, 9	syl5 22	. . . . . . . . . 10 ⊢ ((x ∈ On → (x ⊆ A → x ∈ A)) → (x ∈ (On ∖ A) → (((On ∖ A) ∩ x) = ∅ → x ∈ A)))
11	10	imp 277	. . . . . . . . 9 ⊢ (((x ∈ On → (x ⊆ A → x ∈ A)) ∧ x ∈ (On ∖ A)) → (((On ∖ A) ∩ x) = ∅ → x ∈ A))
12	2, 11	mtod 95	. . . . . . . 8 ⊢ (((x ∈ On → (x ⊆ A → x ∈ A)) ∧ x ∈ (On ∖ A)) → ¬ ((On ∖ A) ∩ x) = ∅)
13	12	exp 291	. . . . . . 7 ⊢ ((x ∈ On → (x ⊆ A → x ∈ A)) → (x ∈ (On ∖ A) → ¬ ((On ∖ A) ∩ x) = ∅))
14	13	19.20i 691	. . . . . 6 ⊢ (∀x(x ∈ On → (x ⊆ A → x ∈ A)) → ∀x(x ∈ (On ∖ A) → ¬ ((On ∖ A) ∩ x) = ∅))
15		df-ral 1205	. . . . . 6 ⊢ (∀x ∈ On (x ⊆ A → x ∈ A) ↔ ∀x(x ∈ On → (x ⊆ A → x ∈ A)))
16		df-ral 1205	. . . . . 6 ⊢ (∀x ∈ (On ∖ A) ¬ ((On ∖ A) ∩ x) = ∅ ↔ ∀x(x ∈ (On ∖ A) → ¬ ((On ∖ A) ∩ x) = ∅))
17	14, 15, 16	3imtr4 192	. . . . 5 ⊢ (∀x ∈ On (x ⊆ A → x ∈ A) → ∀x ∈ (On ∖ A) ¬ ((On ∖ A) ∩ x) = ∅)
18		ralnex 1209	. . . . 5 ⊢ (∀x ∈ (On ∖ A) ¬ ((On ∖ A) ∩ x) = ∅ ↔ ¬ ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
19	17, 18	sylib 173	. . . 4 ⊢ (∀x ∈ On (x ⊆ A → x ∈ A) → ¬ ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
20		ssdif0 1748	. . . . . 6 ⊢ (On ⊆ A ↔ (On ∖ A) = ∅)
21	20	negbii 162	. . . . 5 ⊢ (¬ On ⊆ A ↔ ¬ (On ∖ A) = ∅)
22		difss 1596	. . . . . 6 ⊢ (On ∖ A) ⊆ On
23		ordon 2238	. . . . . . 7 ⊢ Ord On
24		tz7.5 2220	. . . . . . 7 ⊢ ((Ord On ∧ ((On ∖ A) ⊆ On ∧ ¬ (On ∖ A) = ∅)) → ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
25	23, 24	mpan 518	. . . . . 6 ⊢ (((On ∖ A) ⊆ On ∧ ¬ (On ∖ A) = ∅) → ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
26	22, 25	mpan 518	. . . . 5 ⊢ (¬ (On ∖ A) = ∅ → ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
27	21, 26	sylbi 174	. . . 4 ⊢ (¬ On ⊆ A → ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
28	19, 27	nsyl2 103	. . 3 ⊢ (∀x ∈ On (x ⊆ A → x ∈ A) → On ⊆ A)
29	28	anim2i 270	. 2 ⊢ ((A ⊆ On ∧ ∀x ∈ On (x ⊆ A → x ∈ A)) → (A ⊆ On ∧ On ⊆ A))
30		eqss 1516	. 2 ⊢ (A = On ↔ (A ⊆ On ∧ On ⊆ A))
31	29, 30	sylibr 175	1 ⊢ ((A ⊆ On ∧ ∀x ∈ On (x ⊆ A → x ∈ A)) → A = On)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∖ cdif 1484   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  Ord word 2198  Oncon0 2199

This theorem is referenced by:  tfis 2245

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

metamath.org