Theorem ordsssuc2 6486

Description: An ordinal subset of an ordinal number belongs to its successor. (Contributed by NM, 1-Feb-2005.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
ordsssuc2	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))

Proof of Theorem ordsssuc2

Step	Hyp	Ref	Expression
1		elong 6403	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	biimprd 248	. . . 4 ⊢ (𝐴 ∈ V → (Ord 𝐴 → 𝐴 ∈ On))
3	2	anim1d 610	. . 3 ⊢ (𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On)))
4		onsssuc 6485	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
5	3, 4	syl6 35	. 2 ⊢ (𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)))
6		annim 403	. . . . 5 ⊢ ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) ↔ ¬ (𝐵 ∈ On → 𝐴 ∈ V))
7		ssexg 5341	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ On) → 𝐴 ∈ V)
8	7	ex 412	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
9		elex 3509	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝐵 → 𝐴 ∈ V)
10	9	a1d 25	. . . . . 6 ⊢ (𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
11	8, 10	pm5.21ni 377	. . . . 5 ⊢ (¬ (𝐵 ∈ On → 𝐴 ∈ V) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
12	6, 11	sylbi 217	. . . 4 ⊢ ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
13	12	expcom 413	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ On → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)))
14	13	adantld 490	. 2 ⊢ (¬ 𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)))
15	5, 14	pm2.61i 182	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  Ord word 6394  Oncon0 6395  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-suc 6401

This theorem is referenced by:  ordunisuc2  7881