Theorem cofsslt 27866

Description: If every element of 𝐴 is bounded above by some element of 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Note - we will often use the term "cofinal" to denote that every element of 𝐴 is bounded above by some element of 𝐵. Similarly, we will use the term "coinitial" to denote that every element of 𝐴 is bounded below by some element of 𝐵. (Contributed by Scott Fenton, 24-Sep-2024.)

Assertion

Ref	Expression
cofsslt	⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem cofsslt

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ 𝒫 No )
2		ssltex2 27733	. . 3 ⊢ (𝐵 <<s 𝐶 → 𝐶 ∈ V)
3	2	3ad2ant3 1135	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
4	1	elpwid 4568	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No )
5		ssltss2 27735	. . 3 ⊢ (𝐵 <<s 𝐶 → 𝐶 ⊆ No )
6	5	3ad2ant3 1135	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No )
7		breq1 5105	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝑦 ↔ 𝑎 ≤s 𝑦))
8	7	rexbidv 3157	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑎 ≤s 𝑦))
9		simp12 1205	. . . . 5 ⊢ (((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦)
10		simp2 1137	. . . . 5 ⊢ (((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) → 𝑎 ∈ 𝐴)
11	8, 9, 10	rspcdva 3586	. . . 4 ⊢ (((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) → ∃𝑦 ∈ 𝐵 𝑎 ≤s 𝑦)
12		breq2 5106	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 ≤s 𝑦 ↔ 𝑎 ≤s 𝑏))
13	12	cbvrexvw 3214	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑎 ≤s 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑎 ≤s 𝑏)
14	11, 13	sylib 218	. . 3 ⊢ (((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) → ∃𝑏 ∈ 𝐵 𝑎 ≤s 𝑏)
15		simpl11 1249	. . . . . 6 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝐴 ∈ 𝒫 No )
16	15	elpwid 4568	. . . . 5 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝐴 ⊆ No )
17		simpl2 1193	. . . . 5 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑎 ∈ 𝐴)
18	16, 17	sseldd 3944	. . . 4 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑎 ∈ No )
19		simpl13 1251	. . . . . 6 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝐵 <<s 𝐶)
20		ssltss1 27734	. . . . . 6 ⊢ (𝐵 <<s 𝐶 → 𝐵 ⊆ No )
21	19, 20	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝐵 ⊆ No )
22		simprl 770	. . . . 5 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑏 ∈ 𝐵)
23	21, 22	sseldd 3944	. . . 4 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑏 ∈ No )
24	19, 5	syl 17	. . . . 5 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝐶 ⊆ No )
25		simpl3 1194	. . . . 5 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑐 ∈ 𝐶)
26	24, 25	sseldd 3944	. . . 4 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑐 ∈ No )
27		simprr 772	. . . 4 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑎 ≤s 𝑏)
28	19, 22, 25	ssltsepcd 27740	. . . 4 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑏 <s 𝑐)
29	18, 23, 26, 27, 28	slelttrd 27706	. . 3 ⊢ ((((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 ∈ 𝐵 ∧ 𝑎 ≤s 𝑏)) → 𝑎 <s 𝑐)
30	14, 29	rexlimddv 3140	. 2 ⊢ (((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) ∧ 𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶) → 𝑎 <s 𝑐)
31	1, 3, 4, 6, 30	ssltd 27737	1 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3444   ⊆ wss 3911  𝒫 cpw 4559   class class class wbr 5102   No csur 27584   <s cslt 27585   ≤s csle 27689   <<s csslt 27726

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

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-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-1o 8411  df-2o 8412  df-no 27587  df-slt 27588  df-sle 27690  df-sslt 27727

This theorem is referenced by:  cofcut1  27868  cofcut2  27870