Theorem tfi 3113

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 included in 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:   x,A

Proof of Theorem tfi

Step	Hyp	Ref	Expression
1		eldifn 2152	. . . . . . . . 9 ⊢ (x ∈ (On ∖ A) → ¬ x ∈ A)
2	1	adantl 388	. . . . . . . 8 ⊢ (((x ∈ On → (x ⊆ A → x ∈ A)) ⋀ x ∈ (On ∖ A)) → ¬ x ∈ A)
3		difin0ss 2321	. . . . . . . . . . . . 13 ⊢ (((On ∖ A) ∩ x) = ∅ → (x ⊆ On → x ⊆ A))
4		onsst 2979	. . . . . . . . . . . . 13 ⊢ (x ∈ On → x ⊆ On)
5	3, 4	syl5com 52	. . . . . . . . . . . 12 ⊢ (x ∈ On → (((On ∖ A) ∩ x) = ∅ → x ⊆ A))
6	5	imim1d 28	. . . . . . . . . . 11 ⊢ (x ∈ On → ((x ⊆ A → x ∈ A) → (((On ∖ A) ∩ x) = ∅ → x ∈ A)))
7	6	a2i 9	. . . . . . . . . 10 ⊢ ((x ∈ On → (x ⊆ A → x ∈ A)) → (x ∈ On → (((On ∖ A) ∩ x) = ∅ → x ∈ A)))
8		eldifi 2151	. . . . . . . . . 10 ⊢ (x ∈ (On ∖ A) → x ∈ On)
9	7, 8	syl5 21	. . . . . . . . 9 ⊢ ((x ∈ On → (x ⊆ A → x ∈ A)) → (x ∈ (On ∖ A) → (((On ∖ A) ∩ x) = ∅ → x ∈ A)))
10	9	imp 350	. . . . . . . 8 ⊢ (((x ∈ On → (x ⊆ A → x ∈ A)) ⋀ x ∈ (On ∖ A)) → (((On ∖ A) ∩ x) = ∅ → x ∈ A))
11	2, 10	mtod 108	. . . . . . 7 ⊢ (((x ∈ On → (x ⊆ A → x ∈ A)) ⋀ x ∈ (On ∖ A)) → ¬ ((On ∖ A) ∩ x) = ∅)
12	11	ex 373	. . . . . 6 ⊢ ((x ∈ On → (x ⊆ A → x ∈ A)) → (x ∈ (On ∖ A) → ¬ ((On ∖ A) ∩ x) = ∅))
13	12	r19.20i2 1694	. . . . 5 ⊢ (∀x ∈ On (x ⊆ A → x ∈ A) → ∀x ∈ (On ∖ A) ¬ ((On ∖ A) ∩ x) = ∅)
14		ralnex 1644	. . . . 5 ⊢ (∀x ∈ (On ∖ A) ¬ ((On ∖ A) ∩ x) = ∅ ↔ ¬ ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
15	13, 14	sylib 198	. . . 4 ⊢ (∀x ∈ On (x ⊆ A → x ∈ A) → ¬ ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
16		ssdif0 2316	. . . . . 6 ⊢ (On ⊆ A ↔ (On ∖ A) = ∅)
17	16	necon3bbii 1588	. . . . 5 ⊢ (¬ On ⊆ A ↔ (On ∖ A) ≠ ∅)
18		ordon 2974	. . . . . 6 ⊢ Ord On
19		difss 2156	. . . . . 6 ⊢ (On ∖ A) ⊆ On
20		tz7.5 2956	. . . . . 6 ⊢ ((Ord On ⋀ (On ∖ A) ⊆ On ⋀ (On ∖ A) ≠ ∅) → ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
21	18, 19, 20	mp3an12 903	. . . . 5 ⊢ ((On ∖ A) ≠ ∅ → ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
22	17, 21	sylbi 199	. . . 4 ⊢ (¬ On ⊆ A → ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
23	15, 22	nsyl2 118	. . 3 ⊢ (∀x ∈ On (x ⊆ A → x ∈ A) → On ⊆ A)
24	23	anim2i 335	. 2 ⊢ ((A ⊆ On ⋀ ∀x ∈ On (x ⊆ A → x ∈ A)) → (A ⊆ On ⋀ On ⊆ A))
25		eqss 2066	. 2 ⊢ (A = On ↔ (A ⊆ On ⋀ On ⊆ A))
26	24, 25	sylibr 200	1 ⊢ ((A ⊆ On ⋀ ∀x ∈ On (x ⊆ A → x ∈ A)) → A = On)

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   = wceq 953   ∈ wcel 955   ≠ wne 1576  ∀wral 1636  ∃wrex 1637   ∖ cdif 2033   ∩ cin 2035   ⊆ wss 2036  ∅c0 2269  Ord word 2934  Oncon0 2935

This theorem is referenced by:  tfis 3114

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1451  ax-sep 2692  ax-pow 2731  ax-pr 2768  ax-un 2854

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1374  df-mo 1375  df-clab 1456  df-cleq 1461  df-clel 1464  df-ne 1578  df-ral 1640  df-rex 1641  df-v 1802  df-dif 2038  df-un 2039  df-in 2040  df-ss 2042  df-nul 2270  df-pw 2391  df-sn 2401  df-pr 2402  df-tp 2404  df-op 2405  df-uni 2493  df-br 2609  df-opab 2656  df-tr 2670  df-eprel 2818  df-po 2828  df-so 2838  df-fr 2904  df-we 2921  df-ord 2938  df-on 2939

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