Theorem cofon1 8642

Description: Cofinality theorem for ordinals. If 𝐴 is cofinal with 𝐵 and the upper bound of 𝐴 dominates 𝐵, then their upper bounds are equal. Compare with cofcut1 28010 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)

Hypotheses

Ref	Expression
cofon1.1	⊢ (𝜑 → 𝐴 ∈ 𝒫 On)
cofon1.2	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
cofon1.3	⊢ (𝜑 → 𝐵 ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})

Assertion

Ref	Expression
cofon1	⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} = ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})

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

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

Proof of Theorem cofon1

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3962	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝐵 ⊆ 𝑤 ↔ 𝐵 ⊆ 𝑧))
2	1	cbvrabv 3424	. . . 4 ⊢ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} = {𝑧 ∈ On ∣ 𝐵 ⊆ 𝑧}
3		sseq1 3961	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦))
4	3	rexbidv 3186	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦 ↔ ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦))
5		cofon1.2	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
6	5	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
7		simprr 782	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → 𝑎 ∈ 𝐴)
8	4, 6, 7	rspcdva 3582	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → ∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦)
9		sseq2 3962	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑎 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑏))
10	9	cbvrexvw 3241	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏)
11	8, 10	sylib 220	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏)
12		simprl 780	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → 𝐵 ⊆ 𝑧)
13	12	sselda 3936	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑧)
14		cofon1.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ 𝒫 On)
15	14	elpwid 4564	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ On)
16	15	ad3antrrr 740	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝐴 ⊆ On)
17		simplrr 787	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐴)
18	16, 17	sseldd 3937	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ On)
19		simpllr 785	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝑧 ∈ On)
20		ontr2 6394	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ On ∧ 𝑧 ∈ On) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑧) → 𝑎 ∈ 𝑧))
21	18, 19, 20	syl2anc 593	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑧) → 𝑎 ∈ 𝑧))
22	13, 21	mpan2d 704	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → (𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑧))
23	22	rexlimdva 3163	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → (∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑧))
24	11, 23	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → 𝑎 ∈ 𝑧)
25	24	expr 460	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝐵 ⊆ 𝑧) → (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑧))
26	25	ssrdv 3942	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝐵 ⊆ 𝑧) → 𝐴 ⊆ 𝑧)
27	26	ex 416	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐵 ⊆ 𝑧 → 𝐴 ⊆ 𝑧))
28	27	ss2rabdv 4028	. . . 4 ⊢ (𝜑 → {𝑧 ∈ On ∣ 𝐵 ⊆ 𝑧} ⊆ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
29	2, 28	eqsstrid 3974	. . 3 ⊢ (𝜑 → {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
30		intss 4927	. . 3 ⊢ ({𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ⊆ ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})
31	29, 30	syl 17	. 2 ⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ⊆ ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})
32		sseq2 3962	. . . 4 ⊢ (𝑤 = ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} → (𝐵 ⊆ 𝑤 ↔ 𝐵 ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧}))
33		ssorduni 7762	. . . . . . . . 9 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
34	15, 33	syl 17	. . . . . . . 8 ⊢ (𝜑 → Ord ∪ 𝐴)
35		ordsuc 7794	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ Ord suc ∪ 𝐴)
36	34, 35	sylib 220	. . . . . . 7 ⊢ (𝜑 → Ord suc ∪ 𝐴)
37	14	uniexd 7725	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
38		sucexg 7788	. . . . . . . . 9 ⊢ (∪ 𝐴 ∈ V → suc ∪ 𝐴 ∈ V)
39	37, 38	syl 17	. . . . . . . 8 ⊢ (𝜑 → suc ∪ 𝐴 ∈ V)
40		elong 6354	. . . . . . . 8 ⊢ (suc ∪ 𝐴 ∈ V → (suc ∪ 𝐴 ∈ On ↔ Ord suc ∪ 𝐴))
41	39, 40	syl 17	. . . . . . 7 ⊢ (𝜑 → (suc ∪ 𝐴 ∈ On ↔ Ord suc ∪ 𝐴))
42	36, 41	mpbird 259	. . . . . 6 ⊢ (𝜑 → suc ∪ 𝐴 ∈ On)
43		onsucuni 7808	. . . . . . 7 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
44	15, 43	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ suc ∪ 𝐴)
45		sseq2 3962	. . . . . . 7 ⊢ (𝑧 = suc ∪ 𝐴 → (𝐴 ⊆ 𝑧 ↔ 𝐴 ⊆ suc ∪ 𝐴))
46	45	rspcev 3581	. . . . . 6 ⊢ ((suc ∪ 𝐴 ∈ On ∧ 𝐴 ⊆ suc ∪ 𝐴) → ∃𝑧 ∈ On 𝐴 ⊆ 𝑧)
47	42, 44, 46	syl2anc 593	. . . . 5 ⊢ (𝜑 → ∃𝑧 ∈ On 𝐴 ⊆ 𝑧)
48		onintrab2 7780	. . . . 5 ⊢ (∃𝑧 ∈ On 𝐴 ⊆ 𝑧 ↔ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ∈ On)
49	47, 48	sylib 220	. . . 4 ⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ∈ On)
50		cofon1.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
51	32, 49, 50	elrabd 3652	. . 3 ⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ∈ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})
52		intss1 4921	. . 3 ⊢ (∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ∈ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} → ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
53	51, 52	syl 17	. 2 ⊢ (𝜑 → ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
54	31, 53	eqssd 3953	1 ⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} = ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  {crab 3414  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4555  ∪ cuni 4865  ∩ cint 4905  Ord word 6345  Oncon0 6346  suc csuc 6348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-ord 6349  df-on 6350  df-suc 6352

This theorem is referenced by:  cofon2  8643