Theorem cofon1 8654

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 28113 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 3963	. . . . 5 ⊢ (𝑤 = 𝑧 → (𝐵 ⊆ 𝑤 ↔ 𝐵 ⊆ 𝑧))
2	1	cbvrabv 3426	. . . 4 ⊢ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} = {𝑧 ∈ On ∣ 𝐵 ⊆ 𝑧}
3		sseq1 3962	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑥 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑦))
4	3	rexbidv 3189	. . . . . . . . . . 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 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ 𝑦)
7		simprr 784	. . . . . . . . . . 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 3963	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑎 ⊆ 𝑦 ↔ 𝑎 ⊆ 𝑏))
10	9	cbvrexvw 3244	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ 𝐵 𝑎 ⊆ 𝑦 ↔ ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏)
11	8, 10	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → ∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏)
12		simprl 782	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → 𝐵 ⊆ 𝑧)
13	12	sselda 3937	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝑧)
14		cofon1.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ 𝒫 On)
15	14	elpwid 4571	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ On)
16	15	ad3antrrr 742	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝐴 ⊆ On)
17		simplrr 789	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ 𝐴)
18	16, 17	sseldd 3938	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝑎 ∈ On)
19		simpllr 787	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → 𝑧 ∈ On)
20		ontr2 6409	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ On ∧ 𝑧 ∈ On) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑧) → 𝑎 ∈ 𝑧))
21	18, 19, 20	syl2anc 595	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → ((𝑎 ⊆ 𝑏 ∧ 𝑏 ∈ 𝑧) → 𝑎 ∈ 𝑧))
22	13, 21	mpan2d 706	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) ∧ 𝑏 ∈ 𝐵) → (𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑧))
23	22	rexlimdva 3166	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → (∃𝑏 ∈ 𝐵 𝑎 ⊆ 𝑏 → 𝑎 ∈ 𝑧))
24	11, 23	mpd 16	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝐵 ⊆ 𝑧 ∧ 𝑎 ∈ 𝐴)) → 𝑎 ∈ 𝑧)
25	24	expr 461	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝐵 ⊆ 𝑧) → (𝑎 ∈ 𝐴 → 𝑎 ∈ 𝑧))
26	25	ssrdv 3943	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝐵 ⊆ 𝑧) → 𝐴 ⊆ 𝑧)
27	26	ex 417	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐵 ⊆ 𝑧 → 𝐴 ⊆ 𝑧))
28	27	ss2rabdv 4029	. . . 4 ⊢ (𝜑 → {𝑧 ∈ On ∣ 𝐵 ⊆ 𝑧} ⊆ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
29	2, 28	eqsstrid 3975	. . 3 ⊢ (𝜑 → {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
30		intss 4934	. . 3 ⊢ ({𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} ⊆ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ⊆ ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})
31	29, 30	syl 18	. 2 ⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ⊆ ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})
32		sseq2 3963	. . . 4 ⊢ (𝑤 = ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} → (𝐵 ⊆ 𝑤 ↔ 𝐵 ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧}))
33		ssorduni 7774	. . . . . . . . 9 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
34	15, 33	syl 18	. . . . . . . 8 ⊢ (𝜑 → Ord ∪ 𝐴)
35		ordsuc 7806	. . . . . . . 8 ⊢ (Ord ∪ 𝐴 ↔ Ord suc ∪ 𝐴)
36	34, 35	sylib 221	. . . . . . 7 ⊢ (𝜑 → Ord suc ∪ 𝐴)
37	14	uniexd 7740	. . . . . . . . 9 ⊢ (𝜑 → ∪ 𝐴 ∈ V)
38		sucexg 7800	. . . . . . . . 9 ⊢ (∪ 𝐴 ∈ V → suc ∪ 𝐴 ∈ V)
39	37, 38	syl 18	. . . . . . . 8 ⊢ (𝜑 → suc ∪ 𝐴 ∈ V)
40		elong 6368	. . . . . . . 8 ⊢ (suc ∪ 𝐴 ∈ V → (suc ∪ 𝐴 ∈ On ↔ Ord suc ∪ 𝐴))
41	39, 40	syl 18	. . . . . . 7 ⊢ (𝜑 → (suc ∪ 𝐴 ∈ On ↔ Ord suc ∪ 𝐴))
42	36, 41	mpbird 260	. . . . . 6 ⊢ (𝜑 → suc ∪ 𝐴 ∈ On)
43		onsucuni 7820	. . . . . . 7 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
44	15, 43	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ suc ∪ 𝐴)
45		sseq2 3963	. . . . . . 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 595	. . . . 5 ⊢ (𝜑 → ∃𝑧 ∈ On 𝐴 ⊆ 𝑧)
48		onintrab2 7792	. . . . 5 ⊢ (∃𝑧 ∈ On 𝐴 ⊆ 𝑧 ↔ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ∈ On)
49	47, 48	sylib 221	. . . 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 4928	. . 3 ⊢ (∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} ∈ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} → ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
53	51, 52	syl 18	. 2 ⊢ (𝜑 → ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤} ⊆ ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧})
54	31, 53	eqssd 3954	1 ⊢ (𝜑 → ∩ {𝑧 ∈ On ∣ 𝐴 ⊆ 𝑧} = ∩ {𝑤 ∈ On ∣ 𝐵 ⊆ 𝑤})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ⊆ wss 3905  𝒫 cpw 4562  ∪ cuni 4872  ∩ cint 4912  Ord word 6359  Oncon0 6360  suc csuc 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364  df-suc 6366

This theorem is referenced by:  cofon2  8655