Theorem cofon2 8671

Description: Cofinality theorem for ordinals. If 𝐴 and 𝐵 are mutually cofinal, then their upper bounds agree. Compare cofcut2 27793 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 4002	. . . . . . . 8 ⊢ (𝑧 = 𝑏 → (𝑧 ⊆ 𝑤 ↔ 𝑏 ⊆ 𝑤))
4	3	rexbidv 3172	. . . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∀𝑧 ∈ 𝐵 ∃𝑤 ∈ 𝐴 𝑧 ⊆ 𝑤)
7		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
8	4, 6, 7	rspcdva 3607	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤)
9		sseq2 4003	. . . . . . 7 ⊢ (𝑤 = 𝑐 → (𝑏 ⊆ 𝑤 ↔ 𝑏 ⊆ 𝑐))
10	9	cbvrexvw 3229	. . . . . 6 ⊢ (∃𝑤 ∈ 𝐴 𝑏 ⊆ 𝑤 ↔ ∃𝑐 ∈ 𝐴 𝑏 ⊆ 𝑐)
11	8, 10	sylib 217	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ∃𝑐 ∈ 𝐴 𝑏 ⊆ 𝑐)
12		ssintub 4963	. . . . . . . . 9 ⊢ 𝐴 ⊆ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}
13	12	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 ⊆ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
14	13	sselda 3977	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → 𝑐 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
15		cofon2.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ 𝒫 On)
16	15	elpwid 4606	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ⊆ On)
17	16	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → 𝐵 ⊆ On)
18		simplr 766	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → 𝑏 ∈ 𝐵)
19	17, 18	sseldd 3978	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → 𝑏 ∈ On)
20	1	elpwid 4606	. . . . . . . . . . . . . 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 6365	. . . . . . . . . . . . 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 257	. . . . . . . . . . 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 4003	. . . . . . . . . . . 12 ⊢ (𝑎 = suc ∪ 𝐴 → (𝐴 ⊆ 𝑎 ↔ 𝐴 ⊆ suc ∪ 𝐴))
33	32	rspcev 3606	. . . . . . . . . . 11 ⊢ ((suc ∪ 𝐴 ∈ On ∧ 𝐴 ⊆ suc ∪ 𝐴) → ∃𝑎 ∈ On 𝐴 ⊆ 𝑎)
34	29, 31, 33	syl2anc 583	. . . . . . . . . 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 723	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} ∈ On)
38		ontr2 6404	. . . . . . . 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 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → ((𝑏 ⊆ 𝑐 ∧ 𝑐 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}) → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}))
40	14, 39	mpan2d 691	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑐 ∈ 𝐴) → (𝑏 ⊆ 𝑐 → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}))
41	40	rexlimdva 3149	. . . . 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 412	. . 3 ⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎}))
44	43	ssrdv 3983	. 2 ⊢ (𝜑 → 𝐵 ⊆ ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎})
45	1, 2, 44	cofon1 8670	1 ⊢ (𝜑 → ∩ {𝑎 ∈ On ∣ 𝐴 ⊆ 𝑎} = ∩ {𝑏 ∈ On ∣ 𝐵 ⊆ 𝑏})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3055  ∃wrex 3064  {crab 3426  Vcvv 3468   ⊆ wss 3943  𝒫 cpw 4597  ∪ cuni 4902  ∩ cint 4943  Ord word 6356  Oncon0 6357  suc csuc 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6360  df-on 6361  df-suc 6363

This theorem is referenced by:  naddunif  8691