Theorem coflton 8727

Description: Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Compare cofsslt 27970 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)

Hypotheses

Ref	Expression
coflton.1	⊢ (𝜑 → 𝐴 ⊆ On)
coflton.2	⊢ (𝜑 → 𝐵 ⊆ On)
coflton.3	⊢ (𝜑 → 𝐶 ⊆ On)
coflton.4	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
coflton.5	⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤)

Assertion

Ref	Expression
coflton	⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 𝑎 ∈ 𝑐)

Distinct variable groups:   𝐴,𝑐   𝑥,𝐴   𝑥,𝐵,𝑦   𝑧,𝐵   𝑤,𝐶,𝑧   𝑎,𝑐,𝜑   𝑥,𝑎,𝑦   𝑤,𝑐

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑦,𝑧,𝑤,𝑎)   𝐵(𝑤,𝑎,𝑐)   𝐶(𝑥,𝑦,𝑎,𝑐)

Proof of Theorem coflton

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 4034	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦))
2	1	rexbidv 3185	. . . . . 6 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦))
3		coflton.4	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
4	3	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
5		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
6	2, 4, 5	rspcdva 3636	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦)
7	6	adantrr 716	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦)
8		sseq2 4035	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑏))
9	8	cbvrexvw 3244	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏)
10	7, 9	sylib 218	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏)
11		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
12		simplrr 777	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑐 ∈ 𝐶)
13		coflton.5	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤)
14	13	ad2antrr 725	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤)
15		elequ1 2115	. . . . . . 7 ⊢ (𝑧 = 𝑏 → (𝑧 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤))
16		elequ2 2123	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (𝑏 ∈ 𝑤 ↔ 𝑏 ∈ 𝑐))
17	15, 16	rspc2va 3647	. . . . . 6 ⊢ (((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤) → 𝑏 ∈ 𝑐)
18	11, 12, 14, 17	syl21anc 837	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑐)
19		coflton.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ On)
20	19	sselda 4008	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ On)
21	20	adantrr 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ On)
22		coflton.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ On)
23	22	sselda 4008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ On)
24	23	adantrl 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → 𝑐 ∈ On)
25	24	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑐 ∈ On)
26		ontr2 6442	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑐 ∈ On) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑐) → 𝑎 ∈ 𝑐))
27	21, 25, 26	syl2an2r 684	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑐) → 𝑎 ∈ 𝑐))
28	18, 27	mpan2d 693	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → (𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑐))
29	28	rexlimdva 3161	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → (∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑐))
30	10, 29	mpd 15	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ 𝑐)
31	30	ralrimivva 3208	1 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 𝑎 ∈ 𝑐)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  Oncon0 6395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399

This theorem is referenced by: (None)