Theorem cofsslt 33743

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 1137	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ 𝒫 No )
2		ssltex2 33637	. . 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 4509	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No )
5		ssltss2 33639	. . 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 5043	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝑦 ↔ 𝑎 ≤s 𝑦))
8	7	rexbidv 3208	. . . . 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 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 3531	. . . 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 5044	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 ≤s 𝑦 ↔ 𝑎 ≤s 𝑏))
13	12	cbvrexvw 3351	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑎 ≤s 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑎 ≤s 𝑏)
14	11, 13	sylib 221	. . 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 4509	. . . . 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 3888	. . . 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 33638	. . . . . 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 3888	. . . 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 3888	. . . 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	ssltsepcd 33643	. . . 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 33619	. . 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 3202	. 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 33641	1 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1088   ∈ wcel 2114  ∀wral 3054  ∃wrex 3055  Vcvv 3400   ⊆ wss 3853  𝒫 cpw 4498   class class class wbr 5040   No csur 33498   <s cslt 33499   ≤s csle 33602   <<s csslt 33630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-csb 3801  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-tr 5147  df-id 5439  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-ord 6185  df-on 6186  df-suc 6188  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-1o 8143  df-2o 8144  df-no 33501  df-slt 33502  df-sle 33603  df-sslt 33631

This theorem is referenced by:  cofcut1  33745  cofcut2  33746