Theorem cofsslt 34015

Description: If every element of 𝐴 is bounded 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 1134	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ 𝒫 No )
2		ssltex2 33909	. . 3 ⊢ (𝐵 <<s 𝐶 → 𝐶 ∈ V)
3	2	3ad2ant3 1133	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
4	1	elpwid 4541	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No )
5		ssltss2 33911	. . 3 ⊢ (𝐵 <<s 𝐶 → 𝐶 ⊆ No )
6	5	3ad2ant3 1133	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No )
7		breq1 5073	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝑦 ↔ 𝑎 ≤s 𝑦))
8	7	rexbidv 3225	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑎 ≤s 𝑦))
9		simp12 1202	. . . . 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 1135	. . . . 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 3554	. . . 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 5074	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 ≤s 𝑦 ↔ 𝑎 ≤s 𝑏))
13	12	cbvrexvw 3373	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑎 ≤s 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑎 ≤s 𝑏)
14	11, 13	sylib 217	. . 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 1246	. . . . . 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 4541	. . . . 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 1190	. . . . 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 3918	. . . 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 1248	. . . . . 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 33910	. . . . . 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 767	. . . . 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 3918	. . . 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 1191	. . . . 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 3918	. . . 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 769	. . . 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 33915	. . . 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 33891	. . 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 3219	. 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 33913	1 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530   class class class wbr 5070   No csur 33770   <s cslt 33771   ≤s csle 33874   <<s csslt 33902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-1o 8267  df-2o 8268  df-no 33773  df-slt 33774  df-sle 33875  df-sslt 33903

This theorem is referenced by:  cofcut1  34017  cofcut2  34018