Theorem onmindif 2312

Description: When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass.

Assertion

Ref	Expression
onmindif	⊢ ((A ⊆ On ∧ B ∈ On) → B ∈ ∩(A ∖ suc B))

Proof of Theorem onmindif

Step	Hyp	Ref	Expression
1		ontri1 2232	. . . . . . . . . . 11 ⊢ ((x ∈ On ∧ B ∈ On) → (x ⊆ B ↔ ¬ B ∈ x))
2		onsssuc 2311	. . . . . . . . . . 11 ⊢ ((x ∈ On ∧ B ∈ On) → (x ⊆ B ↔ x ∈ suc B))
3	1, 2	bitr3d 408	. . . . . . . . . 10 ⊢ ((x ∈ On ∧ B ∈ On) → (¬ B ∈ x ↔ x ∈ suc B))
4	3	bicon1d 405	. . . . . . . . 9 ⊢ ((x ∈ On ∧ B ∈ On) → (¬ x ∈ suc B ↔ B ∈ x))
5		ssel2 1503	. . . . . . . . 9 ⊢ ((A ⊆ On ∧ x ∈ A) → x ∈ On)
6	4, 5	sylan 343	. . . . . . . 8 ⊢ (((A ⊆ On ∧ x ∈ A) ∧ B ∈ On) → (¬ x ∈ suc B ↔ B ∈ x))
7	6	biimpd 135	. . . . . . 7 ⊢ (((A ⊆ On ∧ x ∈ A) ∧ B ∈ On) → (¬ x ∈ suc B → B ∈ x))
8	7	exp31 293	. . . . . 6 ⊢ (A ⊆ On → (x ∈ A → (B ∈ On → (¬ x ∈ suc B → B ∈ x))))
9	8	com23 32	. . . . 5 ⊢ (A ⊆ On → (B ∈ On → (x ∈ A → (¬ x ∈ suc B → B ∈ x))))
10	9	imp4b 283	. . . 4 ⊢ ((A ⊆ On ∧ B ∈ On) → ((x ∈ A ∧ ¬ x ∈ suc B) → B ∈ x))
11		eldif 1496	. . . 4 ⊢ (x ∈ (A ∖ suc B) ↔ (x ∈ A ∧ ¬ x ∈ suc B))
12	10, 11	syl5ib 181	. . 3 ⊢ ((A ⊆ On ∧ B ∈ On) → (x ∈ (A ∖ suc B) → B ∈ x))
13	12	r19.21aiv 1259	. 2 ⊢ ((A ⊆ On ∧ B ∈ On) → ∀x ∈ (A ∖ suc B)B ∈ x)
14		elintg 1973	. . 3 ⊢ (B ∈ On → (B ∈ ∩(A ∖ suc B) ↔ ∀x ∈ (A ∖ suc B)B ∈ x))
15	14	adantl 305	. 2 ⊢ ((A ⊆ On ∧ B ∈ On) → (B ∈ ∩(A ∖ suc B) ↔ ∀x ∈ (A ∖ suc B)B ∈ x))
16	13, 15	mpbird 171	1 ⊢ ((A ⊆ On ∧ B ∈ On) → B ∈ ∩(A ∖ suc B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196   ∈ wcel 1092  ∀wral 1201   ∖ cdif 1484   ⊆ wss 1487  ∩cint 1965  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  unblem3 3433

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-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-op 1815  df-uni 1920  df-int 1966  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  df-suc 2205

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