Theorem sorpssint 6947

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 4492	. . . . . 6 ⊢ (𝑢 ∈ 𝑌 → ∩ 𝑌 ⊆ 𝑢)
2	1	3ad2ant2 1083	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 ⊆ 𝑢)
3		sorpssi 6943	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
4	3	anassrs 680	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ 𝑣 ∈ 𝑌) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
5		ax-1 6	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑣 → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
6		sspss 3706	. . . . . . . . . . 11 ⊢ (𝑣 ⊆ 𝑢 ↔ (𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢))
7		orel1 397	. . . . . . . . . . . 12 ⊢ (¬ 𝑣 ⊊ 𝑢 → ((𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢) → 𝑣 = 𝑢))
8		eqimss2 3658	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑢 → 𝑢 ⊆ 𝑣)
9	7, 8	syl6com 37	. . . . . . . . . . 11 ⊢ ((𝑣 ⊊ 𝑢 ∨ 𝑣 = 𝑢) → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
10	6, 9	sylbi 207	. . . . . . . . . 10 ⊢ (𝑣 ⊆ 𝑢 → (¬ 𝑣 ⊊ 𝑢 → 𝑢 ⊆ 𝑣))
11	5, 10	jaoi 394	. . . . . . . . 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 2962	. . . . . . 7 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) → (∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 → ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣))
14	13	3impia 1261	. . . . . 6 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣)
15		ssint 4493	. . . . . 6 ⊢ (𝑢 ⊆ ∩ 𝑌 ↔ ∀𝑣 ∈ 𝑌 𝑢 ⊆ 𝑣)
16	14, 15	sylibr 224	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → 𝑢 ⊆ ∩ 𝑌)
17	2, 16	eqssd 3620	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 = 𝑢)
18		simp2 1062	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → 𝑢 ∈ 𝑌)
19	17, 18	eqeltrd 2701	. . 3 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢) → ∩ 𝑌 ∈ 𝑌)
20	19	rexlimdv3a 3033	. 2 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 → ∩ 𝑌 ∈ 𝑌))
21		intss1 4492	. . . . 5 ⊢ (𝑣 ∈ 𝑌 → ∩ 𝑌 ⊆ 𝑣)
22		ssnpss 3710	. . . . 5 ⊢ (∩ 𝑌 ⊆ 𝑣 → ¬ 𝑣 ⊊ ∩ 𝑌)
23	21, 22	syl 17	. . . 4 ⊢ (𝑣 ∈ 𝑌 → ¬ 𝑣 ⊊ ∩ 𝑌)
24	23	rgen 2922	. . 3 ⊢ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌
25		psseq2 3695	. . . . . 6 ⊢ (𝑢 = ∩ 𝑌 → (𝑣 ⊊ 𝑢 ↔ 𝑣 ⊊ ∩ 𝑌))
26	25	notbid 308	. . . . 5 ⊢ (𝑢 = ∩ 𝑌 → (¬ 𝑣 ⊊ 𝑢 ↔ ¬ 𝑣 ⊊ ∩ 𝑌))
27	26	ralbidv 2986	. . . 4 ⊢ (𝑢 = ∩ 𝑌 → (∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 ↔ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌))
28	27	rspcev 3309	. . 3 ⊢ ((∩ 𝑌 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ ∩ 𝑌) → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢)
29	24, 28	mpan2 707	. 2 ⊢ (∩ 𝑌 ∈ 𝑌 → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢)
30	20, 29	impbid1 215	1 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑣 ⊊ 𝑢 ↔ ∩ 𝑌 ∈ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   ⊊ wpss 3575  ∩ cint 4475   Or wor 5034   [⊊] crpss 6936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-br 4654  df-opab 4713  df-so 5036  df-xp 5120  df-rel 5121  df-rpss 6937

This theorem is referenced by:  fin2i2  9140  isfin2-2  9141