Theorem ordsssuc2 6482

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 6396	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ∈ On ↔ Ord 𝐴))
2	1	biimprd 251	. . . 4 ⊢ (𝐴 ∈ V → (Ord 𝐴 → 𝐴 ∈ On))
3	2	anim1d 623	. . 3 ⊢ (𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ∈ On ∧ 𝐵 ∈ On)))
4		onsssuc 6481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
5	3, 4	syl6 36	. 2 ⊢ (𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)))
6		annim 409	. . . . 5 ⊢ ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) ↔ ¬ (𝐵 ∈ On → 𝐴 ∈ V))
7		ssexg 5288	. . . . . . 7 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ On) → 𝐴 ∈ V)
8	7	ex 418	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
9		elex 3475	. . . . . . 7 ⊢ (𝐴 ∈ suc 𝐵 → 𝐴 ∈ V)
10	9	a1d 26	. . . . . 6 ⊢ (𝐴 ∈ suc 𝐵 → (𝐵 ∈ On → 𝐴 ∈ V))
11	8, 10	pm5.21ni 380	. . . . 5 ⊢ (¬ (𝐵 ∈ On → 𝐴 ∈ V) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
12	6, 11	sylbi 220	. . . 4 ⊢ ((𝐵 ∈ On ∧ ¬ 𝐴 ∈ V) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))
13	12	expcom 419	. . 3 ⊢ (¬ 𝐴 ∈ V → (𝐵 ∈ On → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)))
14	13	adantld 496	. 2 ⊢ (¬ 𝐴 ∈ V → ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵)))
15	5, 14	pm2.61i 184	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ 𝐴 ∈ suc 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900  Ord word 6387  Oncon0 6388  suc csuc 6390

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392  df-suc 6394

This theorem is used by:  ordunisuc2  7889