Theorem tfi 4566

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.

(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

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 2453	. . . . . . 7 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)))
2		imdi 249	. . . . . . . 8 ⊢ ((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ↔ ((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
3	2	albii 1463	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ↔ ∀𝑥((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
4	1, 3	bitri 183	. . . . . 6 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
5		dfss2 3136	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
6	5	imbi2i 225	. . . . . . . . 9 ⊢ ((𝑥 ∈ On → 𝑥 ⊆ 𝐴) ↔ (𝑥 ∈ On → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
7		19.21v 1866	. . . . . . . . 9 ⊢ (∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) ↔ (𝑥 ∈ On → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
8	6, 7	bitr4i 186	. . . . . . . 8 ⊢ ((𝑥 ∈ On → 𝑥 ⊆ 𝐴) ↔ ∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
9	8	imbi1i 237	. . . . . . 7 ⊢ (((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) ↔ (∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
10	9	albii 1463	. . . . . 6 ⊢ (∀𝑥((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) ↔ ∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
11	4, 10	bitri 183	. . . . 5 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
12		simpl 108	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝑥)
13		tron 4367	. . . . . . . . . . . . . 14 ⊢ Tr On
14		dftr2 4089	. . . . . . . . . . . . . 14 ⊢ (Tr On ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On))
15	13, 14	mpbi 144	. . . . . . . . . . . . 13 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
16	15	spi 1529	. . . . . . . . . . . 12 ⊢ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
17	16	spi 1529	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
18	12, 17	jca 304	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → (𝑦 ∈ 𝑥 ∧ 𝑦 ∈ On))
19	18	imim1i 60	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ On) → 𝑦 ∈ 𝐴) → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴))
20		impexp 261	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ On) → 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)))
21		impexp 261	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ 𝐴)))
22		bi2.04 247	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ 𝐴)) ↔ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
23	21, 22	bitri 183	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
24	19, 20, 23	3imtr3i 199	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
25	24	alimi 1448	. . . . . . 7 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → ∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
26	25	imim1i 60	. . . . . 6 ⊢ ((∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) → (∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
27	26	alimi 1448	. . . . 5 ⊢ (∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) → ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
28	11, 27	sylbi 120	. . . 4 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
29	28	adantl 275	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
30		sbim 1946	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝑥 ∈ On → [𝑦 / 𝑥]𝑥 ∈ 𝐴))
31		clelsb1 2275	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥]𝑥 ∈ On ↔ 𝑦 ∈ On)
32		clelsb1 2275	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
33	31, 32	imbi12i 238	. . . . . . . . . 10 ⊢ (([𝑦 / 𝑥]𝑥 ∈ On → [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝑦 ∈ On → 𝑦 ∈ 𝐴))
34	30, 33	bitri 183	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) ↔ (𝑦 ∈ On → 𝑦 ∈ 𝐴))
35	34	ralbii 2476	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ On → 𝑦 ∈ 𝐴))
36		df-ral 2453	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ On → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)))
37	35, 36	bitri 183	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)))
38	37	imbi1i 237	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) ↔ (∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
39	38	albii 1463	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) ↔ ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
40		ax-setind 4521	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) → ∀𝑥(𝑥 ∈ On → 𝑥 ∈ 𝐴))
41	39, 40	sylbir 134	. . . 4 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) → ∀𝑥(𝑥 ∈ On → 𝑥 ∈ 𝐴))
42		dfss2 3136	. . . 4 ⊢ (On ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ On → 𝑥 ∈ 𝐴))
43	41, 42	sylibr 133	. . 3 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) → On ⊆ 𝐴)
44	29, 43	syl 14	. 2 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → On ⊆ 𝐴)
45		eqss 3162	. . 3 ⊢ (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
46	45	biimpri 132	. 2 ⊢ ((𝐴 ⊆ On ∧ On ⊆ 𝐴) → 𝐴 = On)
47	44, 46	syldan 280	1 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1346   = wceq 1348  [wsb 1755   ∈ wcel 2141  ∀wral 2448   ⊆ wss 3121  Tr wtr 4087  Oncon0 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353

This theorem is referenced by:  tfis  4567