Theorem tfi 3121

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.

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 2159	. . . . . . . . 9 ⊢ (x ∈ (On ∖ A) → ¬ x ∈ A)
2	1	adantl 388	. . . . . . . 8 ⊢ (((x ∈ On → (x ⊆ A → x ∈ A)) ⋀ x ∈ (On ∖ A)) → ¬ x ∈ A)
3		difin0ss 2328	. . . . . . . . . . . . 13 ⊢ (((On ∖ A) ∩ x) = ∅ → (x ⊆ On → x ⊆ A))
4		onsst 2987	. . . . . . . . . . . . 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 2158	. . . . . . . . . 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 1700	. . . . 5 ⊢ (∀x ∈ On (x ⊆ A → x ∈ A) → ∀x ∈ (On ∖ A) ¬ ((On ∖ A) ∩ x) = ∅)
14		ralnex 1650	. . . . 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 2323	. . . . . 6 ⊢ (On ⊆ A ↔ (On ∖ A) = ∅)
17	16	necon3bbii 1594	. . . . 5 ⊢ (¬ On ⊆ A ↔ (On ∖ A) ≠ ∅)
18		ordon 2982	. . . . . 6 ⊢ Ord On
19		difss 2163	. . . . . 6 ⊢ (On ∖ A) ⊆ On
20		tz7.5 2964	. . . . . 6 ⊢ ((Ord On ⋀ (On ∖ A) ⊆ On ⋀ (On ∖ A) ≠ ∅) → ∃x ∈ (On ∖ A)((On ∖ A) ∩ x) = ∅)
21	18, 19, 20	mp3an12 904	. . . . 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 2073	. 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 954   ∈ wcel 956   ≠ wne 1582  ∀wral 1642  ∃wrex 1643   ∖ cdif 2040   ∩ cin 2042   ⊆ wss 2043  ∅c0 2276  Ord word 2942  Oncon0 2943

This theorem is referenced by:  tfis 3122

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947

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