Theorem cofonr 8667

Description: Inverse cofinality law for ordinals. Contrast with cofcutr 28173 for surreals. (Contributed by Scott Fenton, 20-Jan-2025.)

Hypotheses

Ref	Expression
cofonr.1	⊢ (𝜑 → 𝐴 ∈ On)
cofonr.2	⊢ (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})

Assertion

Ref	Expression
cofonr	⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)

Distinct variable groups:   𝑥,𝐴   𝑧,𝐴   𝑥,𝑋   𝑧,𝑋   𝜑,𝑥,𝑦   𝜑,𝑧,𝑦

Allowed substitution hints:   𝐴(𝑦)   𝑋(𝑦)

Proof of Theorem cofonr

Step	Hyp	Ref	Expression
1		cofonr.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ On)
2		onss 7791	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
3	1, 2	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ On)
4	3	sselda 3938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
5		eloni 6375	. . . . . 6 ⊢ (𝑦 ∈ On → Ord 𝑦)
6		ordirr 6383	. . . . . 6 ⊢ (Ord 𝑦 → ¬ 𝑦 ∈ 𝑦)
7	4, 5, 6	3syl 19	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 ∈ 𝑦)
8		cofonr.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
9	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
10	9	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
11	4	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ On)
12		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ⊆ 𝑦)
13		sseq2 3964	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑋 ⊆ 𝑥 ↔ 𝑋 ⊆ 𝑦))
14	13	elrab 3652	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ↔ (𝑦 ∈ On ∧ 𝑋 ⊆ 𝑦))
15	11, 12, 14	sylanbrc 595	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
16		intss1 4930	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ⊆ 𝑦)
17	15, 16	syl 18	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ⊆ 𝑦)
18	10, 17	eqsstrd 3972	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝐴 ⊆ 𝑦)
19		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ 𝐴)
20	18, 19	sseldd 3939	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ 𝑦)
21	7, 20	mtand 828	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑋 ⊆ 𝑦)
22		dfss3 3927	. . . 4 ⊢ (𝑋 ⊆ 𝑦 ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
23	21, 22	sylnib 331	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
24	8, 1	eqeltrrd 2866	. . . . . . . . . 10 ⊢ (𝜑 → ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ∈ On)
25		onintrab2 7803	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ On 𝑋 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ∈ On)
26	24, 25	sylibr 237	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ On 𝑋 ⊆ 𝑥)
27	26	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ On 𝑋 ⊆ 𝑥)
28		onss 7791	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
29	28	adantl 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ On) → 𝑥 ⊆ On)
30		sstr 3946	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ 𝑥 ∧ 𝑥 ⊆ On) → 𝑋 ⊆ On)
31	30	expcom 419	. . . . . . . . . 10 ⊢ (𝑥 ⊆ On → (𝑋 ⊆ 𝑥 → 𝑋 ⊆ On))
32	29, 31	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ On) → (𝑋 ⊆ 𝑥 → 𝑋 ⊆ On))
33	32	rexlimdva 3168	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑥 ∈ On 𝑋 ⊆ 𝑥 → 𝑋 ⊆ On))
34	27, 33	mpd 16	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑋 ⊆ On)
35	34	sselda 3938	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On)
36		ontri1 6400	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
37	4, 35, 36	syl2an2r 698	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑋) → (𝑦 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
38	37	rexbidva 3189	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝑋 ¬ 𝑧 ∈ 𝑦))
39		rexnal 3119	. . . 4 ⊢ (∃𝑧 ∈ 𝑋 ¬ 𝑧 ∈ 𝑦 ↔ ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
40	38, 39	bitrdi 290	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦))
41	23, 40	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)
42	41	ralrimiva 3159	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  {crab 3418   ⊆ wss 3906  ∩ cint 4914  Ord word 6364  Oncon0 6365

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6368  df-on 6369

This theorem is used by:  naddunif  8687