Theorem ordsucuniel 7760

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 7728	. . 3 ⊢ (Ord 𝐵 → Ord ∪ 𝐵)
2		ordelord 6333	. . . 4 ⊢ ((Ord ∪ 𝐵 ∧ 𝐴 ∈ ∪ 𝐵) → Ord 𝐴)
3	2	ex 412	. . 3 ⊢ (Ord ∪ 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴))
4	1, 3	syl 17	. 2 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 → Ord 𝐴))
5		ordelord 6333	. . . 4 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord suc 𝐴)
6		ordsuc 7750	. . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴)
7	5, 6	sylibr 234	. . 3 ⊢ ((Ord 𝐵 ∧ suc 𝐴 ∈ 𝐵) → Ord 𝐴)
8	7	ex 412	. 2 ⊢ (Ord 𝐵 → (suc 𝐴 ∈ 𝐵 → Ord 𝐴))
9		ordsson 7722	. . . . . 6 ⊢ (Ord 𝐵 → 𝐵 ⊆ On)
10		ordunisssuc 6419	. . . . . 6 ⊢ ((𝐵 ⊆ On ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴))
11	9, 10	sylan 580	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ suc 𝐴))
12		ordtri1 6344	. . . . . 6 ⊢ ((Ord ∪ 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵))
13	1, 12	sylan 580	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (∪ 𝐵 ⊆ 𝐴 ↔ ¬ 𝐴 ∈ ∪ 𝐵))
14		ordtri1 6344	. . . . . 6 ⊢ ((Ord 𝐵 ∧ Ord suc 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴 ∈ 𝐵))
15	6, 14	sylan2b 594	. . . . 5 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ⊆ suc 𝐴 ↔ ¬ suc 𝐴 ∈ 𝐵))
16	11, 13, 15	3bitr3d 309	. . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (¬ 𝐴 ∈ ∪ 𝐵 ↔ ¬ suc 𝐴 ∈ 𝐵))
17	16	con4bid 317	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))
18	17	ex 412	. 2 ⊢ (Ord 𝐵 → (Ord 𝐴 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵)))
19	4, 8, 18	pm5.21ndd 379	1 ⊢ (Ord 𝐵 → (𝐴 ∈ ∪ 𝐵 ↔ suc 𝐴 ∈ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2113   ⊆ wss 3898  ∪ cuni 4858  Ord word 6310  Oncon0 6311  suc csuc 6313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-tr 5201  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-ord 6314  df-on 6315  df-suc 6317

This theorem is referenced by:  naddsuc2  8622  dfac12lem1  10042  dfac12lem2  10043