Theorem ordsucuniel 7821

Description: Given an element 𝐴 of the union of an ordinal 𝐵, suc 𝐴 is an element of 𝐵 itself. (Contributed by Scott Fenton, 28-Mar-2012.) (Proof shortened by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
ordsucuniel	⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))

Proof of Theorem ordsucuniel

Step	Hyp	Ref	Expression
1		orduni 7789	. . 3 ⊢ (Ord 𝐵 → Ord ∪ 𝐵)
2		ordelord 6384	. . . 4 ⊢ ((Ord ∪ 𝐵 ∧ 𝐴 ∈ ∪ 𝐵) → Ord 𝐴)
3	2	ex 417	. . 3 ⊢ (Ord ∪ 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴))
4	1, 3	syl 18	. 2 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴))
5		ordelord 6384	. . . 4 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord suc 𝐴)
6		ordsuc 7811	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
7	5, 6	sylibr 237	. . 3 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord 𝐴)
8	7	ex 417	. 2 ⊢ (Ord 𝐵 → (suc 𝐴 ∈ 𝐵 → Ord 𝐴))
9		ordsson 7783	. . . . . 6 ⊢ (Ord 𝐵 → 𝐵 ⊆ On)
10		ordunisssuc 6471	. . . . . 6 ⊢ ((𝐵 ⊆ On ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴))
11	9, 10	sylan 591	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴))
12		ordtri1 6396	. . . . . 6 ⊢ ((Ord ∪ 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵))
13	1, 12	sylan 591	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵))
14		ordtri1 6396	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴 ∈ 𝐵))
15	6, 14	sylan2b 605	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴 ∈ 𝐵))
16	11, 13, 15	3bitr3d 312	. . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (¬ 𝐴 ∈ ∪ 𝐵 ↔ ¬ suc 𝐴 ∈ 𝐵))
17	16	con4bid 320	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
18	17	ex 417	. 2 ⊢ (Ord 𝐵 → (Ord 𝐴 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)))
19	4, 8, 18	pm5.21ndd 382	1 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2143   ⊆ wss 3906  ∪ cuni 4873  Ord word 6361  Oncon0 6362  suc csuc 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366  df-suc 6368

This theorem is referenced by:  naddsuc2  8689  dfac12lem1  10128  dfac12lem2  10129