Theorem onmindif 6457

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 4049	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵))
2		ssel2 3903	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
3		ontri1 6397	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝑥))
4		onsssuc 6455	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵))
5	3, 4	bitr3d 284	. . . . . . . . . 10 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝐵 ∈ 𝑥 ↔ 𝑥 ∈ suc 𝐵))
6	5	con1bid 358	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥))
7	2, 6	sylan 592	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥))
8	7	biimpd 232	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥))
9	8	exp31 425	. . . . . 6 ⊢ (𝐴 ⊆ On → (𝑥 ∈ 𝐴 → (𝐵 ∈ On → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥))))
10	9	com23 87	. . . . 5 ⊢ (𝐴 ⊆ On → (𝐵 ∈ On → (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥))))
11	10	imp4b 427	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵) → 𝐵 ∈ 𝑥))
12	1, 11	biimtrid 245	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝑥 ∈ (𝐴 ∖ suc 𝐵) → 𝐵 ∈ 𝑥))
13	12	ralrimiv 3153	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥)
14		elintg 4914	. . 3 ⊢ (𝐵 ∈ On → (𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥))
15	14	adantl 487	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → (𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥))
16	13, 15	mpbird 260	1 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → 𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895   ∖ cdif 4045  ∩ cint 4906  Oncon0 6362  suc csuc 6364

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-suc 6368

This theorem is used by:  unblem3  9290  fin23lem26  10407  inaex  45375