Theorem cofslts 27926

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
cofslts	⊢ ((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 cofslts

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

Step	Hyp	Ref	Expression
1		simp1 1137	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ 𝒫 No )
2		sltsex2 27772	. . 3 ⊢ (𝐵 <<s 𝐶 → 𝐶 ∈ V)
3	2	3ad2ant3 1136	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
4	1	elpwid 4565	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No )
5		sltsss2 27774	. . 3 ⊢ (𝐵 <<s 𝐶 → 𝐶 ⊆ No )
6	5	3ad2ant3 1136	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No )
7		breq1 5103	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝑦 ↔ 𝑎 ≤s 𝑦))
8	7	rexbidv 3162	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑎 ≤s 𝑦))
9		simp12 1206	. . . . 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 1138	. . . . 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 3579	. . . 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 5104	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 ≤s 𝑦 ↔ 𝑎 ≤s 𝑏))
13	12	cbvrexvw 3217	. . . 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 1250	. . . . . 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 4565	. . . . 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 1194	. . . . 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 3936	. . . 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 1252	. . . . . 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		sltsss1 27773	. . . . . 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 771	. . . . 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 3936	. . . 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 1195	. . . . 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 3936	. . . 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 773	. . . 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	sltssepcd 27780	. . . 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	leltstrd 27745	. . 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 3145	. 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	sltsd 27776	1 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  𝒫 cpw 4556   class class class wbr 5100   No csur 27619   <s clts 27620   ≤s cles 27724   <<s cslts 27765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-1o 8407  df-2o 8408  df-no 27622  df-lts 27623  df-les 27725  df-slts 27766

This theorem is referenced by:  cofcut1  27928  cofcut2  27930