Theorem cofslts 28337

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 1154	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ∈ 𝒫 No)
2		sltsex2 28183	. . 3 ⊢ (𝐵 <<s 𝐶 → 𝐶 ∈ V)
3	2	3ad2ant3 1153	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐶 ∈ V)
4	1	elpwid 4569	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 ⊆ No)
5		sltsss2 28185	. . 3 ⊢ (𝐵 <<s 𝐶 → 𝐶 ⊆ No)
6	5	3ad2ant3 1153	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐶 ⊆ No)
7		breq1 5109	. . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝑦 ↔ 𝑎 ≤s 𝑦))
8	7	rexbidv 3190	. . . . 5 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑎 ≤s 𝑦))
9		simp12 1223	. . . . 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 1155	. . . . 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 3581	. . . 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 5110	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 ≤s 𝑦 ↔ 𝑎 ≤s 𝑏))
13	12	cbvrexvw 3245	. . . 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 1267	. . . . . 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 4569	. . . . 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 1211	. . . . 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 3914	. . . 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 1269	. . . . . 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 28184	. . . . . 6 ⊢ (𝐵 <<s 𝐶 → 𝐵 ⊆ No)
21	19, 20	syl 18	. . . . 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 783	. . . . 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 3914	. . . 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 18	. . . . 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 1212	. . . . 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 3914	. . . 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 785	. . . 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 28191	. . . 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 28155	. . 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 3173	. 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 28187	1 ⊢ ((𝐴 ∈ 𝒫 No ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ≤s 𝑦 ∧ 𝐵 <<s 𝐶) → 𝐴 <<s 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  𝒫 cpw 4560   class class class wbr 5106  Nocsur 28030   <s clts 28031   ≤s cles 28134   <<s cslts 28176

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-les 28135  df-slts 28177

This theorem is used by:  cofcut1  28339  cofcut2  28341