Theorem sorpssuni 7307

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 7304	. . . . . . . . . 10 ⊢ (( [⊊] Or 𝑌 ∧ (𝑢 ∈ 𝑌 ∧ 𝑣 ∈ 𝑌)) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
2	1	anassrs 468	. . . . . . . . 9 ⊢ ((( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) ∧ 𝑣 ∈ 𝑌) → (𝑢 ⊆ 𝑣 ∨ 𝑣 ⊆ 𝑢))
3		sspss 3992	. . . . . . . . . . 11 ⊢ (𝑢 ⊆ 𝑣 ↔ (𝑢 ⊊ 𝑣 ∨ 𝑢 = 𝑣))
4		orel1 881	. . . . . . . . . . . 12 ⊢ (¬ 𝑢 ⊊ 𝑣 → ((𝑢 ⊊ 𝑣 ∨ 𝑢 = 𝑣) → 𝑢 = 𝑣))
5		eqimss2 3940	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → 𝑣 ⊆ 𝑢)
6	4, 5	syl6com 37	. . . . . . . . . . 11 ⊢ ((𝑢 ⊊ 𝑣 ∨ 𝑢 = 𝑣) → (¬ 𝑢 ⊊ 𝑣 → 𝑣 ⊆ 𝑢))
7	3, 6	sylbi 218	. . . . . . . . . 10 ⊢ (𝑢 ⊆ 𝑣 → (¬ 𝑢 ⊊ 𝑣 → 𝑣 ⊆ 𝑢))
8		ax-1 6	. . . . . . . . . 10 ⊢ (𝑣 ⊆ 𝑢 → (¬ 𝑢 ⊊ 𝑣 → 𝑣 ⊆ 𝑢))
9	7, 8	jaoi 852	. . . . . . . . 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 3142	. . . . . . 7 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌) → (∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 → ∀𝑣 ∈ 𝑌 𝑣 ⊆ 𝑢))
12	11	3impia 1108	. . . . . 6 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → ∀𝑣 ∈ 𝑌 𝑣 ⊆ 𝑢)
13		unissb 4770	. . . . . 6 ⊢ (∪ 𝑌 ⊆ 𝑢 ↔ ∀𝑣 ∈ 𝑌 𝑣 ⊆ 𝑢)
14	12, 13	sylibr 235	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → ∪ 𝑌 ⊆ 𝑢)
15		elssuni 4768	. . . . . 6 ⊢ (𝑢 ∈ 𝑌 → 𝑢 ⊆ ∪ 𝑌)
16	15	3ad2ant2 1125	. . . . 5 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → 𝑢 ⊆ ∪ 𝑌)
17	14, 16	eqssd 3901	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → ∪ 𝑌 = 𝑢)
18		simp2 1128	. . . 4 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → 𝑢 ∈ 𝑌)
19	17, 18	eqeltrd 2881	. . 3 ⊢ (( [⊊] Or 𝑌 ∧ 𝑢 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣) → ∪ 𝑌 ∈ 𝑌)
20	19	rexlimdv3a 3246	. 2 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 → ∪ 𝑌 ∈ 𝑌))
21		elssuni 4768	. . . . 5 ⊢ (𝑣 ∈ 𝑌 → 𝑣 ⊆ ∪ 𝑌)
22		ssnpss 3996	. . . . 5 ⊢ (𝑣 ⊆ ∪ 𝑌 → ¬ ∪ 𝑌 ⊊ 𝑣)
23	21, 22	syl 17	. . . 4 ⊢ (𝑣 ∈ 𝑌 → ¬ ∪ 𝑌 ⊊ 𝑣)
24	23	rgen 3113	. . 3 ⊢ ∀𝑣 ∈ 𝑌 ¬ ∪ 𝑌 ⊊ 𝑣
25		psseq1 3980	. . . . . 6 ⊢ (𝑢 = ∪ 𝑌 → (𝑢 ⊊ 𝑣 ↔ ∪ 𝑌 ⊊ 𝑣))
26	25	notbid 319	. . . . 5 ⊢ (𝑢 = ∪ 𝑌 → (¬ 𝑢 ⊊ 𝑣 ↔ ¬ ∪ 𝑌 ⊊ 𝑣))
27	26	ralbidv 3162	. . . 4 ⊢ (𝑢 = ∪ 𝑌 → (∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 ↔ ∀𝑣 ∈ 𝑌 ¬ ∪ 𝑌 ⊊ 𝑣))
28	27	rspcev 3554	. . 3 ⊢ ((∪ 𝑌 ∈ 𝑌 ∧ ∀𝑣 ∈ 𝑌 ¬ ∪ 𝑌 ⊊ 𝑣) → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣)
29	24, 28	mpan2 687	. 2 ⊢ (∪ 𝑌 ∈ 𝑌 → ∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣)
30	20, 29	impbid1 226	1 ⊢ ( [⊊] Or 𝑌 → (∃𝑢 ∈ 𝑌 ∀𝑣 ∈ 𝑌 ¬ 𝑢 ⊊ 𝑣 ↔ ∪ 𝑌 ∈ 𝑌))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1078   = wceq 1520   ∈ wcel 2079  ∀wral 3103  ∃wrex 3104   ⊆ wss 3854   ⊊ wpss 3855  ∪ cuni 4739   Or wor 5353   [⊊] crpss 7297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-so 5355  df-xp 5441  df-rel 5442  df-rpss 7298

This theorem is referenced by:  fin2i2  9575  isfin2-2  9576  fin12  9670