Theorem sorpssint 7709

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

Assertion

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

Distinct variable group:   𝑢,𝑌,𝑣

Proof of Theorem sorpssint

Step	Hyp	Ref	Expression
1		intss1 4927	. . . . . 6 ⊢ (𝑢 ∈ 𝑌 → ∩ 𝑌 ⊆ 𝑢)
2	1	3ad2ant2 1134	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 ⊆ 𝑢)
3		sorpssi 7705	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
4	3	anassrs 467	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ 𝑣 ∈ 𝑌) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
5		sspss 4065	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝑢 ↔ (𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢))
6		orel1 888	. . . . . . . . . . . 12 ⊢ (¬ 𝑣 ⊊ 𝑢 → ((𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢) → 𝑣 = 𝑢))
7		eqimss2 4006	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑢 → 𝑢 ⊆ 𝑣)
8	6, 7	syl6com 37	. . . . . . . . . . 11 ⊢ ((𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢) → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
9	5, 8	sylbi 217	. . . . . . . . . 10 ⊢ (𝑣 ⊆ 𝑢 → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
10	9	jao1i 858	. . . . . . . . 9 ⊢ ((𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢) → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
11	4, 10	syl 17	. . . . . . . 8 ⊢ ((( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ 𝑣 ∈ 𝑌) → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
12	11	ralimdva 3145	. . . . . . 7 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) → (∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 → ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣))
13	12	3impia 1117	. . . . . 6 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣)
14		ssint 4928	. . . . . 6 ⊢ (𝑢 ⊆ ∩ 𝑌 ↔ ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣)
15	13, 14	sylibr 234	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → 𝑢 ⊆ ∩ 𝑌)
16	2, 15	eqssd 3964	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 = 𝑢)
17		simp2 1137	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → 𝑢 ∈ 𝑌)
18	16, 17	eqeltrd 2828	. . 3 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 ∈ 𝑌)
19	18	rexlimdv3a 3138	. 2 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 → ∩ 𝑌 ∈ 𝑌))
20		intss1 4927	. . . . 5 ⊢ (𝑣 ∈ 𝑌 → ∩ 𝑌 ⊆ 𝑣)
21		ssnpss 4069	. . . . 5 ⊢ (∩ 𝑌 ⊆ 𝑣 → ¬ 𝑣 ⊊ ∩ 𝑌)
22	20, 21	syl 17	. . . 4 ⊢ (𝑣 ∈ 𝑌 → ¬ 𝑣 ⊊ ∩ 𝑌)
23	22	rgen 3046	. . 3 ⊢ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌
24		psseq2 4054	. . . . . 6 ⊢ (𝑢 = ∩ 𝑌 → (𝑣 ⊊ 𝑢 ↔ 𝑣 ⊊ ∩ 𝑌))
25	24	notbid 318	. . . . 5 ⊢ (𝑢 = ∩ 𝑌 → (¬ 𝑣 ⊊ 𝑢 ↔ ¬ 𝑣 ⊊ ∩ 𝑌))
26	25	ralbidv 3156	. . . 4 ⊢ (𝑢 = ∩ 𝑌 → (∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 ↔ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌))
27	26	rspcev 3588	. . 3 ⊢ ((∩ 𝑌 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌) → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢)
28	23, 27	mpan2 691	. 2 ⊢ (∩ 𝑌 ∈ 𝑌 → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢)
29	19, 28	impbid1 225	1 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 ↔ ∩ 𝑌 ∈ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914   ⊊ wpss 3915  ∩ cint 4910   Or wor 5545   [⊊] crpss 7698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-int 4911  df-br 5108  df-opab 5170  df-so 5547  df-xp 5644  df-rel 5645  df-rpss 7699

This theorem is referenced by:  fin2i2  10271  isfin2-2  10272