Theorem onmindif 6340

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 3893	. . . 4 ⊢ (𝑥 ∈ (𝐴 ∖ suc 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵))
2		ssel2 3912	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
3		ontri1 6285	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝑥))
4		onsssuc 6338	. . . . . . . . . . 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 355	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝐵 ∈ On) → (¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥))
7	2, 6	sylan 579	. . . . . . . 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 419	. . . . . 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 421	. . . 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 3106	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥)
14		elintg 4884	. . 3 ⊢ (𝐵 ∈ On → (𝐵 ∈ ∩ (𝐴 ∖ suc 𝐵) ↔ ∀𝑥 ∈ (𝐴 ∖ suc 𝐵)𝐵 ∈ 𝑥))
15	14	adantl 481	. 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 395   ∈ wcel 2108  ∀wral 3063   ∖ cdif 3880   ⊆ wss 3883  ∩ cint 4876  Oncon0 6251  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-suc 6257

This theorem is referenced by:  unblem3  8998  fin23lem26  10012  inaex  41804