Theorem coflton 8680

Description: Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Compare cofslts 28304 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 3933	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦))
2	1	rexbidv 3186	. . . . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
5		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ 𝐴)
6	2, 4, 5	rspcdva 3577	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦)
7	6	adantrr 730	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦)
8		sseq2 3934	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑏))
9	8	cbvrexvw 3241	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏)
10	7, 9	sylib 221	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏)
11		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
12		simplrr 790	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑐 ∈ 𝐶)
13		coflton.5	. . . . . . 7 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤)
14	13	ad2antrr 739	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤)
15		elequ1 2151	. . . . . . 7 ⊢ (𝑧 = 𝑏 → (𝑧 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤))
16		elequ2 2159	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (𝑏 ∈ 𝑤 ↔ 𝑏 ∈ 𝑐))
17	15, 16	rspc2va 3587	. . . . . 6 ⊢ (((𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐶) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤) → 𝑏 ∈ 𝑐)
18	11, 12, 14, 17	syl21anc 851	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑐)
19		coflton.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ On)
20	19	sselda 3908	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ On)
21	20	adantrr 730	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ On)
22		coflton.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ On)
23	22	sselda 3908	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ On)
24	23	adantrl 729	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → 𝑐 ∈ On)
25	24	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑐 ∈ On)
26		ontr2 6411	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑐 ∈ On) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑐) → 𝑎 ∈ 𝑐))
27	21, 25, 26	syl2an2r 698	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑐) → 𝑎 ∈ 𝑐))
28	18, 27	mpan2d 707	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → (𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑐))
29	28	rexlimdva 3163	. . 3 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → (∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑐))
30	10, 29	mpd 16	. 2 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ 𝑐)
31	30	ralrimivva 3205	1 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 𝑎 ∈ 𝑐)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  Oncon0 6362

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366

This theorem is used by: (None)