Theorem onmindif 6355

Description: When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003.)

Assertion

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

Proof of Theorem onmindif

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3897	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵))
2		ssel2 3916	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
3		ontri1 6300	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝑥))
4		onsssuc 6353	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵))
5	3, 4	bitr3d 280	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 ∈ 𝑥 ↔ 𝑥 ∈ suc 𝐵))
6	5	con1bid 356	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥))
7	2, 6	sylan 580	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥))
8	7	biimpd 228	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥))
9	8	exp31 420	. . . . . 6 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → (𝐵 ∈ On → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥))))
10	9	com23 86	. . . . 5 ⊢ (𝐴 ⊆ On → (𝐵 ∈ On → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥))))
11	10	imp4b 422	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵) → 𝐵 ∈ 𝑥))
12	1, 11	syl5bi 241	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝑥 ∈ (𝐴 ∖ suc 𝐵) → 𝐵 ∈ 𝑥))
13	12	ralrimiv 3102	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥)
14		elintg 4887	. . 3 ⊢ (𝐵 ∈ On → (𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥))
15	14	adantl 482	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥))
16	13, 15	mpbird 256	1 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∀wral 3064   ∖ cdif 3884   ⊆ wss 3887  ∩ cint 4879  Oncon0 6266  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-suc 6272

This theorem is referenced by:  unblem3  9068  fin23lem26  10081  inaex  41915