Theorem coflton 8612

Description: Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and 𝐵 precedes 𝐶, then 𝐴 precedes 𝐶. Compare cofsslt 27866 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 3969	. . . . . . 7 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦))
2	1	rexbidv 3157	. . . . . 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 3586	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦)
7	6	adantrr 717	. . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦)
8		sseq2 3970	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑎 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑏))
9	8	cbvrexvw 3214	. . . 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 726	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐶 𝑧 ∈ 𝑤)
15		elequ1 2116	. . . . . . 7 ⊢ (𝑧 = 𝑏 → (𝑧 ∈ 𝑤 ↔ 𝑏 ∈ 𝑤))
16		elequ2 2124	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (𝑏 ∈ 𝑤 ↔ 𝑏 ∈ 𝑐))
17	15, 16	rspc2va 3597	. . . . . 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 3943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ On)
21	20	adantrr 717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → 𝑎 ∈ On)
22		coflton.3	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ On)
23	22	sselda 3943	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ On)
24	23	adantrl 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) → 𝑐 ∈ On)
25	24	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → 𝑐 ∈ On)
26		ontr2 6368	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑐 ∈ On) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑐) → 𝑎 ∈ 𝑐))
27	21, 25, 26	syl2an2r 685	. . . . 5 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑐) → 𝑎 ∈ 𝑐))
28	18, 27	mpan2d 694	. . . 4 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝐴 ∧ 𝑐 ∈ 𝐶)) ∧ 𝑏 ∈ 𝐵) → (𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑐))
29	28	rexlimdva 3134	. . 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 3178	1 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑐 ∈ 𝐶 𝑎 ∈ 𝑐)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911  Oncon0 6320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-ord 6323  df-on 6324

This theorem is referenced by: (None)