Theorem tfi 4724

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 2533	. . . . . . 7 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)))
2		imdi 250	. . . . . . . 8 ⊢ ((𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ↔ ((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
3	2	albii 1523	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ On → (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) ↔ ∀𝑥((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
4	1, 3	bitri 184	. . . . . 6 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
5		ssalel 3235	. . . . . . . . . 10 ⊢ (𝑥 ⊆ 𝐴 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
6	5	imbi2i 226	. . . . . . . . 9 ⊢ ((𝑥 ∈ On → 𝑥 ⊆ 𝐴) ↔ (𝑥 ∈ On → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
7		19.21v 1926	. . . . . . . . 9 ⊢ (∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) ↔ (𝑥 ∈ On → ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
8	6, 7	bitr4i 187	. . . . . . . 8 ⊢ ((𝑥 ∈ On → 𝑥 ⊆ 𝐴) ↔ ∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
9	8	imbi1i 238	. . . . . . 7 ⊢ (((𝑥 ∈ On → 𝑥 ⊆ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) ↔ (∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
10	9	albii 1523	. . . . . 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 184	. . . . 5 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) ↔ ∀𝑥(∀𝑦(𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
12		simpl 109	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝑥)
13		tron 4522	. . . . . . . . . . . . . 14 ⊢ Tr On
14		dftr2 4226	. . . . . . . . . . . . . 14 ⊢ (Tr On ↔ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On))
15	13, 14	mpbi 145	. . . . . . . . . . . . 13 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
16	15	spi 1589	. . . . . . . . . . . 12 ⊢ ∀𝑥((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
17	16	spi 1589	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
18	12, 17	jca 306	. . . . . . . . . 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 263	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑦 ∈ On) → 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)))
21		impexp 263	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ 𝐴)))
22		bi2.04 248	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝑥 → (𝑥 ∈ On → 𝑦 ∈ 𝐴)) ↔ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
23	21, 22	bitri 184	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
24	19, 20, 23	3imtr3i 200	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
25	24	alimi 1508	. . . . . . 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 1508	. . . . 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 121	. . . 4 ⊢ (∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴) → ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
29	28	adantl 277	. . 3 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → ∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)))
30		sbim 2013	. . . . . . . . . 10 ⊢ ([𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝑥 ∈ On → [𝑦 / 𝑥]𝑥 ∈ 𝐴))
31		clelsb1 2343	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥]𝑥 ∈ On ↔ 𝑦 ∈ On)
32		clelsb1 2343	. . . . . . . . . . 11 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
33	31, 32	imbi12i 239	. . . . . . . . . 10 ⊢ (([𝑦 / 𝑥]𝑥 ∈ On → [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝑦 ∈ On → 𝑦 ∈ 𝐴))
34	30, 33	bitri 184	. . . . . . . . 9 ⊢ ([𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) ↔ (𝑦 ∈ On → 𝑦 ∈ 𝐴))
35	34	ralbii 2556	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) ↔ ∀𝑦 ∈ 𝑥 (𝑦 ∈ On → 𝑦 ∈ 𝐴))
36		df-ral 2533	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 ∈ On → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)))
37	35, 36	bitri 184	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)))
38	37	imbi1i 238	. . . . . 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 1523	. . . . 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 4679	. . . . 5 ⊢ (∀𝑥(∀𝑦 ∈ 𝑥 [𝑦 / 𝑥](𝑥 ∈ On → 𝑥 ∈ 𝐴) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) → ∀𝑥(𝑥 ∈ On → 𝑥 ∈ 𝐴))
41	39, 40	sylbir 135	. . . 4 ⊢ (∀𝑥(∀𝑦(𝑦 ∈ 𝑥 → (𝑦 ∈ On → 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐴)) → ∀𝑥(𝑥 ∈ On → 𝑥 ∈ 𝐴))
42		ssalel 3235	. . . 4 ⊢ (On ⊆ 𝐴 ↔ ∀𝑥(𝑥 ∈ On → 𝑥 ∈ 𝐴))
43	41, 42	sylibr 134	. . 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 3263	. . 3 ⊢ (𝐴 = On ↔ (𝐴 ⊆ On ∧ On ⊆ 𝐴))
46	45	biimpri 133	. 2 ⊢ ((𝐴 ⊆ On ∧ On ⊆ 𝐴) → 𝐴 = On)
47	44, 46	syldan 282	1 ⊢ ((𝐴 ⊆ On ∧ ∀𝑥 ∈ On (𝑥 ⊆ 𝐴 → 𝑥 ∈ 𝐴)) → 𝐴 = On)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1400   = wceq 1402  [wsb 1815   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  Tr wtr 4224  Oncon0 4503

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-in 3226  df-ss 3233  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508

This theorem is referenced by:  tfis  4725