Theorem cofonr 8730

Description: Inverse cofinality law for ordinals. Contrast with cofcutr 27976 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 7820	. . . . . . . 8 ⊢ (𝐴 ∈ On → 𝐴 ⊆ On)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ On)
4	3	sselda 4008	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
5		eloni 6405	. . . . . 6 ⊢ (𝑦 ∈ On → Ord 𝑦)
6		ordirr 6413	. . . . . 6 ⊢ (Ord 𝑦 → ¬ 𝑦 ∈ 𝑦)
7	4, 5, 6	3syl 18	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 ∈ 𝑦)
8		cofonr.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
9	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
10	9	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
11	4	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ On)
12		simpr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ⊆ 𝑦)
13		sseq2 4035	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑋 ⊆ 𝑥 ↔ 𝑋 ⊆ 𝑦))
14	13	elrab 3708	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ↔ (𝑦 ∈ On ∧ 𝑋 ⊆ 𝑦))
15	11, 12, 14	sylanbrc 582	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
16		intss1 4987	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} → ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ⊆ 𝑦)
17	15, 16	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ⊆ 𝑦)
18	10, 17	eqsstrd 4047	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝐴 ⊆ 𝑦)
19		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ 𝐴)
20	18, 19	sseldd 4009	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ 𝑦)
21	7, 20	mtand 815	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑋 ⊆ 𝑦)
22		dfss3 3997	. . . 4 ⊢ (𝑋 ⊆ 𝑦 ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
23	21, 22	sylnib 328	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
24	8, 1	eqeltrrd 2845	. . . . . . . . . 10 ⊢ (𝜑 → ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ∈ On)
25		onintrab2 7833	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ On 𝑋 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ∈ On)
26	24, 25	sylibr 234	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ On 𝑋 ⊆ 𝑥)
27	26	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ On 𝑋 ⊆ 𝑥)
28		onss 7820	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
29	28	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ On) → 𝑥 ⊆ On)
30		sstr 4017	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ 𝑥 ∧ 𝑥 ⊆ On) → 𝑋 ⊆ On)
31	30	expcom 413	. . . . . . . . . 10 ⊢ (𝑥 ⊆ On → (𝑋 ⊆ 𝑥 → 𝑋 ⊆ On))
32	29, 31	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ On) → (𝑋 ⊆ 𝑥 → 𝑋 ⊆ On))
33	32	rexlimdva 3161	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑥 ∈ On 𝑋 ⊆ 𝑥 → 𝑋 ⊆ On))
34	27, 33	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑋 ⊆ On)
35	34	sselda 4008	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On)
36		ontri1 6429	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
37	4, 35, 36	syl2an2r 684	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑋) → (𝑦 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
38	37	rexbidva 3183	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝑋 ¬ 𝑧 ∈ 𝑦))
39		rexnal 3106	. . . 4 ⊢ (∃𝑧 ∈ 𝑋 ¬ 𝑧 ∈ 𝑦 ↔ ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
40	38, 39	bitrdi 287	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦))
41	23, 40	mpbird 257	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)
42	41	ralrimiva 3152	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ⊆ wss 3976  ∩ cint 4970  Ord word 6394  Oncon0 6395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399

This theorem is referenced by:  naddunif  8749