Theorem onsupneqmaxlim0 41958

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 4915	. 2 ⊢ (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 ⊆ ∪ ∪ 𝐴)
2		ssorduni 7762	. . . . 5 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
3		orduniss 6458	. . . . 5 ⊢ (Ord ∪ 𝐴 → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
4	2, 3	syl 17	. . . 4 ⊢ (𝐴 ⊆ On → ∪ ∪ 𝐴 ⊆ ∪ 𝐴)
5	4	biantrud 532	. . 3 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴)))
6		eqss 3996	. . 3 ⊢ (∪ 𝐴 = ∪ ∪ 𝐴 ↔ (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ∧ ∪ ∪ 𝐴 ⊆ ∪ 𝐴))
7	5, 6	bitr4di 288	. 2 ⊢ (𝐴 ⊆ On → (∪ 𝐴 ⊆ ∪ ∪ 𝐴 ↔ ∪ 𝐴 = ∪ ∪ 𝐴))
8	1, 7	imbitrid 243	1 ⊢ (𝐴 ⊆ On → (𝐴 ⊆ ∪ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ⊆ wss 3947  ∪ cuni 4907  Ord word 6360  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365

This theorem is referenced by: (None)