Theorem ordsucun 2333

Description: The successor of the maximum (i.e. union) of two ordinals is the maximum of their successors.

Assertion

Ref	Expression
ordsucun	⊢ ((Ord A ∧ Ord B) → suc (A ∪ B) = (suc A ∪ suc B))

Proof of Theorem ordsucun

Step	Hyp	Ref	Expression
1		ordssun 2330	. . . . . . . . 9 ⊢ ((Ord A ∧ Ord B) → (x ⊆ (A ∪ B) ↔ (x ⊆ A ∨ x ⊆ B)))
2	1	adantl 305	. . . . . . . 8 ⊢ ((x ∈ On ∧ (Ord A ∧ Ord B)) → (x ⊆ (A ∪ B) ↔ (x ⊆ A ∨ x ⊆ B)))
3		ordsssuc 2310	. . . . . . . . 9 ⊢ ((x ∈ On ∧ Ord (A ∪ B)) → (x ⊆ (A ∪ B) ↔ x ∈ suc (A ∪ B)))
4		ordun 2332	. . . . . . . . 9 ⊢ ((Ord A ∧ Ord B) → Ord (A ∪ B))
5	3, 4	sylan2 346	. . . . . . . 8 ⊢ ((x ∈ On ∧ (Ord A ∧ Ord B)) → (x ⊆ (A ∪ B) ↔ x ∈ suc (A ∪ B)))
6		ordsssuc 2310	. . . . . . . . . 10 ⊢ ((x ∈ On ∧ Ord A) → (x ⊆ A ↔ x ∈ suc A))
7	6	adantrr 312	. . . . . . . . 9 ⊢ ((x ∈ On ∧ (Ord A ∧ Ord B)) → (x ⊆ A ↔ x ∈ suc A))
8		ordsssuc 2310	. . . . . . . . . 10 ⊢ ((x ∈ On ∧ Ord B) → (x ⊆ B ↔ x ∈ suc B))
9	8	adantrl 311	. . . . . . . . 9 ⊢ ((x ∈ On ∧ (Ord A ∧ Ord B)) → (x ⊆ B ↔ x ∈ suc B))
10	7, 9	orbi12d 475	. . . . . . . 8 ⊢ ((x ∈ On ∧ (Ord A ∧ Ord B)) → ((x ⊆ A ∨ x ⊆ B) ↔ (x ∈ suc A ∨ x ∈ suc B)))
11	2, 5, 10	3bitr3d 423	. . . . . . 7 ⊢ ((x ∈ On ∧ (Ord A ∧ Ord B)) → (x ∈ suc (A ∪ B) ↔ (x ∈ suc A ∨ x ∈ suc B)))
12		elun 1601	. . . . . . 7 ⊢ (x ∈ (suc A ∪ suc B) ↔ (x ∈ suc A ∨ x ∈ suc B))
13	11, 12	syl6bbr 416	. . . . . 6 ⊢ ((x ∈ On ∧ (Ord A ∧ Ord B)) → (x ∈ suc (A ∪ B) ↔ x ∈ (suc A ∪ suc B)))
14	13	exp 291	. . . . 5 ⊢ (x ∈ On → ((Ord A ∧ Ord B) → (x ∈ suc (A ∪ B) ↔ x ∈ (suc A ∪ suc B))))
15	14	com12 13	. . . 4 ⊢ ((Ord A ∧ Ord B) → (x ∈ On → (x ∈ suc (A ∪ B) ↔ x ∈ (suc A ∪ suc B))))
16	15	pm5.32d 491	. . 3 ⊢ ((Ord A ∧ Ord B) → ((x ∈ On ∧ x ∈ suc (A ∪ B)) ↔ (x ∈ On ∧ x ∈ (suc A ∪ suc B))))
17		ordsuc 2318	. . . . . 6 ⊢ (Ord (A ∪ B) ↔ Ord suc (A ∪ B))
18		ordelon 2222	. . . . . . 7 ⊢ ((Ord suc (A ∪ B) ∧ x ∈ suc (A ∪ B)) → x ∈ On)
19	18	exp 291	. . . . . 6 ⊢ (Ord suc (A ∪ B) → (x ∈ suc (A ∪ B) → x ∈ On))
20	17, 19	sylbi 174	. . . . 5 ⊢ (Ord (A ∪ B) → (x ∈ suc (A ∪ B) → x ∈ On))
21	4, 20	syl 12	. . . 4 ⊢ ((Ord A ∧ Ord B) → (x ∈ suc (A ∪ B) → x ∈ On))
22		pm4.71r 482	. . . 4 ⊢ ((x ∈ suc (A ∪ B) → x ∈ On) ↔ (x ∈ suc (A ∪ B) ↔ (x ∈ On ∧ x ∈ suc (A ∪ B))))
23	21, 22	sylib 173	. . 3 ⊢ ((Ord A ∧ Ord B) → (x ∈ suc (A ∪ B) ↔ (x ∈ On ∧ x ∈ suc (A ∪ B))))
24		ordun 2332	. . . . . 6 ⊢ ((Ord suc A ∧ Ord suc B) → Ord (suc A ∪ suc B))
25		ordelon 2222	. . . . . . 7 ⊢ ((Ord (suc A ∪ suc B) ∧ x ∈ (suc A ∪ suc B)) → x ∈ On)
26	25	exp 291	. . . . . 6 ⊢ (Ord (suc A ∪ suc B) → (x ∈ (suc A ∪ suc B) → x ∈ On))
27	24, 26	syl 12	. . . . 5 ⊢ ((Ord suc A ∧ Ord suc B) → (x ∈ (suc A ∪ suc B) → x ∈ On))
28		ordsuc 2318	. . . . 5 ⊢ (Ord A ↔ Ord suc A)
29		ordsuc 2318	. . . . 5 ⊢ (Ord B ↔ Ord suc B)
30	27, 28, 29	syl2anb 350	. . . 4 ⊢ ((Ord A ∧ Ord B) → (x ∈ (suc A ∪ suc B) → x ∈ On))
31		pm4.71r 482	. . . 4 ⊢ ((x ∈ (suc A ∪ suc B) → x ∈ On) ↔ (x ∈ (suc A ∪ suc B) ↔ (x ∈ On ∧ x ∈ (suc A ∪ suc B))))
32	30, 31	sylib 173	. . 3 ⊢ ((Ord A ∧ Ord B) → (x ∈ (suc A ∪ suc B) ↔ (x ∈ On ∧ x ∈ (suc A ∪ suc B))))
33	16, 23, 32	3bitr4d 424	. 2 ⊢ ((Ord A ∧ Ord B) → (x ∈ suc (A ∪ B) ↔ x ∈ (suc A ∪ suc B)))
34	33	cleqrd 1100	1 ⊢ ((Ord A ∧ Ord B) → suc (A ∪ B) = (suc A ∪ suc B))

Syntax hints:   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ∪ cun 1485   ⊆ wss 1487  Ord word 2198  Oncon0 2199  suc csuc 2201

This theorem is referenced by:  rankpr 3536

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205

metamath.org