Theorem sorpssint 6845

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 4427	. . . . . 6 ⊢ (𝑢 ∈ 𝑌 → ∩ 𝑌 ⊆ 𝑢)
2	1	3ad2ant2 1076	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 ⊆ 𝑢)
3		sorpssi 6841	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
4	3	anassrs 678	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ 𝑣 ∈ 𝑌) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
5		ax-1 6	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑣 → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
6		sspss 3668	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝑢 ↔ (𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢))
7		orel1 396	. . . . . . . . . . . 12 ⊢ (¬ 𝑣 ⊊ 𝑢 → ((𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢) → 𝑣 = 𝑢))
8		eqimss2 3621	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑢 → 𝑢 ⊆ 𝑣)
9	7, 8	syl6com 36	. . . . . . . . . . 11 ⊢ ((𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢) → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
10	6, 9	sylbi 206	. . . . . . . . . 10 ⊢ (𝑣 ⊆ 𝑢 → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
11	5, 10	jaoi 393	. . . . . . . . 9 ⊢ ((𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢) → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
12	4, 11	syl 17	. . . . . . . 8 ⊢ ((( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ 𝑣 ∈ 𝑌) → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
13	12	ralimdva 2945	. . . . . . 7 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) → (∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 → ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣))
14	13	3impia 1253	. . . . . 6 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣)
15		ssint 4428	. . . . . 6 ⊢ (𝑢 ⊆ ∩ 𝑌 ↔ ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣)
16	14, 15	sylibr 223	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → 𝑢 ⊆ ∩ 𝑌)
17	2, 16	eqssd 3585	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 = 𝑢)
18		simp2 1055	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → 𝑢 ∈ 𝑌)
19	17, 18	eqeltrd 2688	. . 3 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 ∈ 𝑌)
20	19	rexlimdv3a 3015	. 2 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 → ∩ 𝑌 ∈ 𝑌))
21		intss1 4427	. . . . 5 ⊢ (𝑣 ∈ 𝑌 → ∩ 𝑌 ⊆ 𝑣)
22		ssnpss 3672	. . . . 5 ⊢ (∩ 𝑌 ⊆ 𝑣 → ¬ 𝑣 ⊊ ∩ 𝑌)
23	21, 22	syl 17	. . . 4 ⊢ (𝑣 ∈ 𝑌 → ¬ 𝑣 ⊊ ∩ 𝑌)
24	23	rgen 2906	. . 3 ⊢ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌
25		psseq2 3657	. . . . . 6 ⊢ (𝑢 = ∩ 𝑌 → (𝑣 ⊊ 𝑢 ↔ 𝑣 ⊊ ∩ 𝑌))
26	25	notbid 307	. . . . 5 ⊢ (𝑢 = ∩ 𝑌 → (¬ 𝑣 ⊊ 𝑢 ↔ ¬ 𝑣 ⊊ ∩ 𝑌))
27	26	ralbidv 2969	. . . 4 ⊢ (𝑢 = ∩ 𝑌 → (∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 ↔ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌))
28	27	rspcev 3282	. . 3 ⊢ ((∩ 𝑌 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌) → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢)
29	24, 28	mpan2 703	. 2 ⊢ (∩ 𝑌 ∈ 𝑌 → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢)
30	20, 29	impbid1 214	1 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 ↔ ∩ 𝑌 ∈ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540   ⊊ wpss 3541  ∩ cint 4410   Or wor 4958   [⊊] crpss 6834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-int 4411  df-br 4584  df-opab 4644  df-so 4960  df-xp 5044  df-rel 5045  df-rpss 6835

This theorem is referenced by:  fin2i2  9023  isfin2-2  9024