Theorem cofon2 8668

Description: Cofinality theorem for ordinals. If 𝐴 and 𝐵 are mutually cofinal, then their upper bounds agree. Compare cofcut2 27398 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)

Hypotheses

Ref	Expression
cofon2.1	⊢ (𝜑 → 𝐴 ∈ 𝒫 On)
cofon2.2	⊢ (𝜑 → 𝐵 ∈ 𝒫 On)
cofon2.3	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
cofon2.4	⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)

Assertion

Ref	Expression
cofon2	⊢ (𝜑 → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} = ∩ {𝑏 ∈ On ∣ 𝐵 ⊆ 𝑏})

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

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

Proof of Theorem cofon2

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cofon2.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝒫 On)
2		cofon2.3	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
3		sseq1 4006	. . . . . . . 8 ⊢ (𝑧 = 𝑏 → (𝑧 ⊆ 𝑤 ↔ 𝑏 ⊆ 𝑤))
4	3	rexbidv 3178	. . . . . . 7 ⊢ (𝑧 = 𝑏 → (∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤))
5		cofon2.4	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
6	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
7		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
8	4, 6, 7	rspcdva 3613	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤)
9		sseq2 4007	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (𝑏 ⊆ 𝑤 ↔ 𝑏 ⊆ 𝑐))
10	9	cbvrexvw 3235	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤 ↔ ∃𝑐 ∈ 𝐴 𝑏 ⊆ 𝑐)
11	8, 10	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑐 ∈ 𝐴 𝑏 ⊆ 𝑐)
12		ssintub 4969	. . . . . . . . 9 ⊢ 𝐴 ⊆ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 ⊆ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
14	13	sselda 3981	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
15		cofon2.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ 𝒫 On)
16	15	elpwid 4610	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
17	16	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → 𝐵 ⊆ On)
18		simplr 767	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → 𝑏 ∈ 𝐵)
19	17, 18	sseldd 3982	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → 𝑏 ∈ On)
20	1	elpwid 4610	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ On)
21		ssorduni 7762	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
22	20, 21	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → Ord ∪ 𝐴)
23		ordsuc 7797	. . . . . . . . . . . . 13 ⊢ (Ord ∪ 𝐴 ↔ Ord suc ∪ 𝐴)
24	22, 23	sylib 217	. . . . . . . . . . . 12 ⊢ (𝜑 → Ord suc ∪ 𝐴)
25	1	uniexd 7728	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
26		sucexg 7789	. . . . . . . . . . . . 13 ⊢ (∪ 𝐴 ∈ V → suc ∪ 𝐴 ∈ V)
27		elong 6369	. . . . . . . . . . . . 13 ⊢ (suc ∪ 𝐴 ∈ V → (suc ∪ 𝐴 ∈ On ↔ Ord suc ∪ 𝐴))
28	25, 26, 27	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (suc ∪ 𝐴 ∈ On ↔ Ord suc ∪ 𝐴))
29	24, 28	mpbird 256	. . . . . . . . . . 11 ⊢ (𝜑 → suc ∪ 𝐴 ∈ On)
30		onsucuni 7812	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
31	20, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ suc ∪ 𝐴)
32		sseq2 4007	. . . . . . . . . . . 12 ⊢ (𝑎 = suc ∪ 𝐴 → (𝐴 ⊆ 𝑎 ↔ 𝐴 ⊆ suc ∪ 𝐴))
33	32	rspcev 3612	. . . . . . . . . . 11 ⊢ ((suc ∪ 𝐴 ∈ On ∧ 𝐴 ⊆ suc ∪ 𝐴) → ∃𝑎 ∈ On 𝐴 ⊆ 𝑎)
34	29, 31, 33	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ∃𝑎 ∈ On 𝐴 ⊆ 𝑎)
35		onintrab2 7781	. . . . . . . . . 10 ⊢ (∃𝑎 ∈ On 𝐴 ⊆ 𝑎 ↔ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} ∈ On)
36	34, 35	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} ∈ On)
37	36	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} ∈ On)
38		ontr2 6408	. . . . . . . 8 ⊢ ((𝑏 ∈ On ∧ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} ∈ On) → ((𝑏 ⊆ 𝑐 ∧ 𝑐 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}) → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}))
39	19, 37, 38	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → ((𝑏 ⊆ 𝑐 ∧ 𝑐 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}) → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}))
40	14, 39	mpan2d 692	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → (𝑏 ⊆ 𝑐 → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}))
41	40	rexlimdva 3155	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (∃𝑐 ∈ 𝐴 𝑏 ⊆ 𝑐 → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}))
42	11, 41	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
43	42	ex 413	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}))
44	43	ssrdv 3987	. 2 ⊢ (𝜑 → 𝐵 ⊆ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
45	1, 2, 44	cofon1 8667	1 ⊢ (𝜑 → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} = ∩ {𝑏 ∈ On ∣ 𝐵 ⊆ 𝑏})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  {crab 3432  Vcvv 3474   ⊆ wss 3947  𝒫 cpw 4601  ∪ cuni 4907  ∩ cint 4949  Ord word 6360  Oncon0 6361  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-suc 6367

This theorem is referenced by:  naddunif  8688