Theorem orduniorsuc 7824

Description: An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.)

Assertion

Ref	Expression
orduniorsuc	⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))

Proof of Theorem orduniorsuc

Step	Hyp	Ref	Expression
1		orduniss 6451	. . . . . 6 ⊢ (Ord 𝐴 → ∪ 𝐴 ⊆ 𝐴)
2		orduni 7786	. . . . . . . 8 ⊢ (Ord 𝐴 → Ord ∪ 𝐴)
3		ordelssne 6378	. . . . . . . 8 ⊢ ((Ord ∪ 𝐴 ∧ Ord 𝐴) → (∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴)))
4	2, 3	mpancom 701	. . . . . . 7 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴)))
5	4	biimprd 251	. . . . . 6 ⊢ (Ord 𝐴 → ((∪ 𝐴 ⊆ 𝐴 ∧ ∪ 𝐴 ≠ 𝐴) → ∪ 𝐴 ∈ 𝐴))
6	1, 5	mpand 708	. . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → ∪ 𝐴 ∈ 𝐴))
7		ordsucss 7812	. . . . 5 ⊢ (Ord 𝐴 → (∪ 𝐴 ∈ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴))
8	6, 7	syld 48	. . . 4 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → suc ∪ 𝐴 ⊆ 𝐴))
9		ordsucuni 7823	. . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ suc ∪ 𝐴)
10	8, 9	jctild 535	. . 3 ⊢ (Ord 𝐴 → (∪ 𝐴 ≠ 𝐴 → (𝐴 ⊆ suc ∪ 𝐴 ∧ suc ∪ 𝐴 ⊆ 𝐴)))
11		df-ne 2956	. . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ¬ 𝐴 = ∪ 𝐴)
12		necom 3008	. . . 4 ⊢ (𝐴 ≠ ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴)
13	11, 12	bitr3i 280	. . 3 ⊢ (¬ 𝐴 = ∪ 𝐴 ↔ ∪ 𝐴 ≠ 𝐴)
14		eqss 3945	. . 3 ⊢ (𝐴 = suc ∪ 𝐴 ↔ (𝐴 ⊆ suc ∪ 𝐴 ∧ suc ∪ 𝐴 ⊆ 𝐴))
15	10, 13, 14	3imtr4g 299	. 2 ⊢ (Ord 𝐴 → (¬ 𝐴 = ∪ 𝐴 → 𝐴 = suc ∪ 𝐴))
16	15	orrd 877	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ∨ 𝐴 = suc ∪ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   ⊆ wss 3898  ∪ cuni 4866  Ord word 6350  suc csuc 6353

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6354  df-on 6355  df-suc 6357

This theorem is used by:  onuniorsuc  7831  oeeulem  8588  cantnfp1lem2  9658  cantnflem1  9668  cnfcom2lem  9680  dfac12lem1  10194  dfac12lem2  10195  ttukeylem3  10561  ttukeylem5  10563  ttukeylem6  10564  ordtoplem  37145  ordcmp  37157  onsucuni3  38210  aomclem5  44003  omlimcl2  44187  onov0suclim  44219  dflim5  44274  onsetreclem3  50722