Theorem onsupneqmaxlim0 43806

Description: If the supremum of a class of ordinals is not in that class, then the supremum is a limit ordinal or empty. (Contributed by RP, 27-Jan-2025.)

Assertion

Ref	Expression
onsupneqmaxlim0	⊢ (𝐴 ⊆ On → (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

Proof of Theorem onsupneqmaxlim0

Step	Hyp	Ref	Expression
1		uniss 4875	. 2 ⊢ (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴)
2		ssorduni 7764	. . . . 5 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		orduniss 6447	. . . . 5 ⊢ (Ord ∪ 𝐴 → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
4	2, 3	syl 17	. . . 4 ⊢ (𝐴 ⊆ On → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
5	4	biantrud 539	. . 3 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴)))
6		eqss 3953	. . 3 ⊢ (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴))
7	5, 6	bitr4di 291	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴))
8	1, 7	imbitrid 246	1 ⊢ (𝐴 ⊆ On → (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ⊆ wss 3906  ∪ cuni 4867  Ord word 6347  Oncon0 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-ord 6351  df-on 6352

This theorem is referenced by: (None)