Theorem onsupeqnmax 43286

Description: Condition when the supremum of a class of ordinals is not the maximum element of that class. (Contributed by RP, 27-Jan-2025.)

Assertion

Ref	Expression
onsupeqnmax	⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (∪ 𝐴 = ∪ ∪ 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴)))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem onsupeqnmax

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ On)
2	1	sselda 3934	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
3		ssel2 3929	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
4	3	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ On)
5		ontri1 6340	. . . . . . . 8 ⊢ ((𝑦 ∈ On ∧ 𝑥 ∈ On) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
6	2, 4, 5	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ⊆ 𝑥 ↔ ¬ 𝑥 ∈ 𝑦))
7	6	ralbidva 3153	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
8	7	rexbidva 3154	. . . . 5 ⊢ (𝐴 ⊆ On → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
9	8	notbid 318	. . . 4 ⊢ (𝐴 ⊆ On → (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦))
10	9	bicomd 223	. . 3 ⊢ (𝐴 ⊆ On → (¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥))
11		dfrex2 3059	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)
12	11	ralbii 3078	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ∀𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)
13		ralnex 3058	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)
14	12, 13	bitri 275	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 ∈ 𝑦)
15		unielid 43258	. . . 4 ⊢ (∪ 𝐴 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
16	15	notbii 320	. . 3 ⊢ (¬ ∪ 𝐴 ∈ 𝐴 ↔ ¬ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 𝑦 ⊆ 𝑥)
17	10, 14, 16	3bitr4g 314	. 2 ⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ ¬ ∪ 𝐴 ∈ 𝐴))
18		onsupnmax 43267	. . 3 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 → ∪ 𝐴 = ∪ ∪ 𝐴))
19	18	pm4.71rd 562	. 2 ⊢ (𝐴 ⊆ On → (¬ ∪ 𝐴 ∈ 𝐴 ↔ (∪ 𝐴 = ∪ ∪ 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴)))
20	17, 19	bitrd 279	1 ⊢ (𝐴 ⊆ On → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝑦 ↔ (∪ 𝐴 = ∪ ∪ 𝐴 ∧ ¬ ∪ 𝐴 ∈ 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3902  ∪ cuni 4859  Oncon0 6306

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

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-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-ord 6309  df-on 6310  df-suc 6312

This theorem is referenced by: (None)