Theorem cofonr 8699

Description: Inverse cofinality law for ordinals. Contrast with cofcutr 27862 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 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ On)
4	3	sselda 3980	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
5		eloni 6382	. . . . . 6 ⊢ (𝑦 ∈ On → Ord 𝑦)
6		ordirr 6390	. . . . . 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 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
10	9	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝐴 = ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
11	4	adantr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ On)
12		simpr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑋 ⊆ 𝑦)
13		sseq2 4006	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑋 ⊆ 𝑥 ↔ 𝑋 ⊆ 𝑦))
14	13	elrab 3682	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ↔ (𝑦 ∈ On ∧ 𝑋 ⊆ 𝑦))
15	11, 12, 14	sylanbrc 581	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥})
16		intss1 4968	. . . . . . . 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 4018	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝐴 ⊆ 𝑦)
19		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ 𝐴)
20	18, 19	sseldd 3981	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑋 ⊆ 𝑦) → 𝑦 ∈ 𝑦)
21	7, 20	mtand 814	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ 𝑋 ⊆ 𝑦)
22		dfss3 3968	. . . 4 ⊢ (𝑋 ⊆ 𝑦 ↔ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
23	21, 22	sylnib 327	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
24	8, 1	eqeltrrd 2829	. . . . . . . . . 10 ⊢ (𝜑 → ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ∈ On)
25		onintrab2 7804	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ On 𝑋 ⊆ 𝑥 ↔ ∩ {𝑥 ∈ On ∣ 𝑋 ⊆ 𝑥} ∈ On)
26	24, 25	sylibr 233	. . . . . . . . 9 ⊢ (𝜑 → ∃𝑥 ∈ On 𝑋 ⊆ 𝑥)
27	26	adantr 479	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑥 ∈ On 𝑋 ⊆ 𝑥)
28		onss 7791	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ On)
29	28	adantl 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ∈ On) → 𝑥 ⊆ On)
30		sstr 3988	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ 𝑥 ∧ 𝑥 ⊆ On) → 𝑋 ⊆ On)
31	30	expcom 412	. . . . . . . . . 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 3151	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑥 ∈ On 𝑋 ⊆ 𝑥 → 𝑋 ⊆ On))
34	27, 33	mpd 15	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑋 ⊆ On)
35	34	sselda 3980	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ On)
36		ontri1 6406	. . . . . 6 ⊢ ((𝑦 ∈ On ∧ 𝑧 ∈ On) → (𝑦 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
37	4, 35, 36	syl2an2r 683	. . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐴) ∧ 𝑧 ∈ 𝑋) → (𝑦 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑦))
38	37	rexbidva 3172	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧 ↔ ∃𝑧 ∈ 𝑋 ¬ 𝑧 ∈ 𝑦))
39		rexnal 3096	. . . 4 ⊢ (∃𝑧 ∈ 𝑋 ¬ 𝑧 ∈ 𝑦 ↔ ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦)
40	38, 39	bitrdi 286	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝑋 𝑧 ∈ 𝑦))
41	23, 40	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)
42	41	ralrimiva 3142	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝑋 𝑦 ⊆ 𝑧)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3057  ∃wrex 3066  {crab 3428   ⊆ wss 3947  ∩ cint 4951  Ord word 6371  Oncon0 6372

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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-int 4952  df-br 5151  df-opab 5213  df-tr 5268  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-ord 6375  df-on 6376

This theorem is referenced by:  naddunif  8718