Theorem sorpssuni 7520

Description: In a chain of sets, a maximal element is the union of the chain. (Contributed by Stefan O'Rear, 2-Nov-2014.)

Assertion

Ref	Expression
sorpssuni	⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 ↔ ∪ 𝑌 ∈ 𝑌))

Distinct variable group:   𝑢,𝑌,𝑣

Proof of Theorem sorpssuni

Step	Hyp	Ref	Expression
1		sorpssi 7517	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
2	1	anassrs 471	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ 𝑣 ∈ 𝑌) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
3		sspss 4014	. . . . . . . . . . 11 ⊢ (𝑢 ⊆ 𝑣 ↔ (𝑢 ⊊ 𝑣 ∨ 𝑢 = 𝑣))
4		orel1 889	. . . . . . . . . . . 12 ⊢ (¬ 𝑢 ⊊ 𝑣 → ((𝑢 ⊊ 𝑣 ∨ 𝑢 = 𝑣) → 𝑢 = 𝑣))
5		eqimss2 3958	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → 𝑣 ⊆ 𝑢)
6	4, 5	syl6com 37	. . . . . . . . . . 11 ⊢ ((𝑢 ⊊ 𝑣 ∨ 𝑢 = 𝑣) → (¬ 𝑢 ⊊ 𝑣 → 𝑣 ⊆ 𝑢))
7	3, 6	sylbi 220	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑣 → (¬ 𝑢 ⊊ 𝑣 → 𝑣 ⊆ 𝑢))
8		ax-1 6	. . . . . . . . . 10 ⊢ (𝑣 ⊆ 𝑢 → (¬ 𝑢 ⊊ 𝑣 → 𝑣 ⊆ 𝑢))
9	7, 8	jaoi 857	. . . . . . . . 9 ⊢ ((𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢) → (¬ 𝑢 ⊊ 𝑣 → 𝑣 ⊆ 𝑢))
10	2, 9	syl 17	. . . . . . . 8 ⊢ ((( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ 𝑣 ∈ 𝑌) → (¬ 𝑢 ⊊ 𝑣 → 𝑣 ⊆ 𝑢))
11	10	ralimdva 3100	. . . . . . 7 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) → (∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 → ∀𝑣 ∈ 𝑌 𝑣 ⊆ 𝑢))
12	11	3impia 1119	. . . . . 6 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → ∀𝑣 ∈ 𝑌 𝑣 ⊆ 𝑢)
13		unissb 4853	. . . . . 6 ⊢ (∪ 𝑌 ⊆ 𝑢 ↔ ∀𝑣 ∈ 𝑌 𝑣 ⊆ 𝑢)
14	12, 13	sylibr 237	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → ∪ 𝑌 ⊆ 𝑢)
15		elssuni 4851	. . . . . 6 ⊢ (𝑢 ∈ 𝑌 → 𝑢 ⊆ ∪ 𝑌)
16	15	3ad2ant2 1136	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → 𝑢 ⊆ ∪ 𝑌)
17	14, 16	eqssd 3918	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → ∪ 𝑌 = 𝑢)
18		simp2 1139	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → 𝑢 ∈ 𝑌)
19	17, 18	eqeltrd 2838	. . 3 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → ∪ 𝑌 ∈ 𝑌)
20	19	rexlimdv3a 3205	. 2 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 → ∪ 𝑌 ∈ 𝑌))
21		elssuni 4851	. . . . 5 ⊢ (𝑣 ∈ 𝑌 → 𝑣 ⊆ ∪ 𝑌)
22		ssnpss 4018	. . . . 5 ⊢ (𝑣 ⊆ ∪ 𝑌 → ¬ ∪ 𝑌 ⊊ 𝑣)
23	21, 22	syl 17	. . . 4 ⊢ (𝑣 ∈ 𝑌 → ¬ ∪ 𝑌 ⊊ 𝑣)
24	23	rgen 3071	. . 3 ⊢ ∀𝑣 ∈ 𝑌 ¬ ∪ 𝑌 ⊊ 𝑣
25		psseq1 4002	. . . . . 6 ⊢ (𝑢 = ∪ 𝑌 → (𝑢 ⊊ 𝑣 ↔ ∪ 𝑌 ⊊ 𝑣))
26	25	notbid 321	. . . . 5 ⊢ (𝑢 = ∪ 𝑌 → (¬ 𝑢 ⊊ 𝑣 ↔ ¬ ∪ 𝑌 ⊊ 𝑣))
27	26	ralbidv 3118	. . . 4 ⊢ (𝑢 = ∪ 𝑌 → (∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 ↔ ∀𝑣 ∈ 𝑌 ¬ ∪ 𝑌 ⊊ 𝑣))
28	27	rspcev 3537	. . 3 ⊢ ((∪ 𝑌 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ ∪ 𝑌 ⊊ 𝑣) → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣)
29	24, 28	mpan2 691	. 2 ⊢ (∪ 𝑌 ∈ 𝑌 → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣)
30	20, 29	impbid1 228	1 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 ↔ ∪ 𝑌 ∈ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 847   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  ∀wral 3061  ∃wrex 3062   ⊆ wss 3866   ⊊ wpss 3867  ∪ cuni 4819   Or wor 5467   [⊊] crpss 7510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-so 5469  df-xp 5557  df-rel 5558  df-rpss 7511

This theorem is referenced by:  fin2i2  9932  isfin2-2  9933  fin12  10027