Theorem soinfdom 35717

Description: A strict order relation on an infinite set dominates that set. (Contributed by BTernaryTau, 14-Jul-2026.)

Assertion

Ref	Expression
soinfdom	⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉 ∧ ω ≼ 𝐴) → 𝐴 ≼ 𝑅)

Proof of Theorem soinfdom

Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		infn0 9294	. . . . . . 7 ⊢ (ω ≼ 𝐴 → 𝐴 ≠ ∅)
2		n0 4300	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦 ∈ 𝐴)
3	1, 2	sylib 221	. . . . . 6 ⊢ (ω ≼ 𝐴 → ∃𝑦 𝑦 ∈ 𝐴)
4	3	adantr 486	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ (𝑅 ∈ 𝑉 ∧ 𝑅 Or 𝐴)) → ∃𝑦 𝑦 ∈ 𝐴)
5		infdifsn 9658	. . . . . . . . 9 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ≈ 𝐴)
6	5	ensymd 9032	. . . . . . . 8 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ (𝐴 ∖ {𝑦}))
7		eldifsn 4748	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝐴 ∖ {𝑦}) ↔ (𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝑦))
8		sotrine 5599	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 Or 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑧 ≠ 𝑦 ↔ (𝑧𝑅𝑦 ∨ 𝑦𝑅𝑧)))
9	8	biimpd 232	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Or 𝐴 ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑧 ≠ 𝑦 → (𝑧𝑅𝑦 ∨ 𝑦𝑅𝑧)))
10	9	ancom2s 663	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Or 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑧 ≠ 𝑦 → (𝑧𝑅𝑦 ∨ 𝑦𝑅𝑧)))
11	10	expr 462	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 Or 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ 𝐴 → (𝑧 ≠ 𝑦 → (𝑧𝑅𝑦 ∨ 𝑦𝑅𝑧))))
12	11	impd 416	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Or 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 ∈ 𝐴 ∧ 𝑧 ≠ 𝑦) → (𝑧𝑅𝑦 ∨ 𝑦𝑅𝑧)))
13	7, 12	biimtrid 245	. . . . . . . . . . . . 13 ⊢ ((𝑅 Or 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ (𝐴 ∖ {𝑦}) → (𝑧𝑅𝑦 ∨ 𝑦𝑅𝑧)))
14		iftrue 4488	. . . . . . . . . . . . . . 15 ⊢ (𝑧𝑅𝑦 → if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) = ⟨𝑧, 𝑦⟩)
15		df-br 5104	. . . . . . . . . . . . . . . 16 ⊢ (𝑧𝑅𝑦 ↔ ⟨𝑧, 𝑦⟩ ∈ 𝑅)
16	15	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (𝑧𝑅𝑦 → ⟨𝑧, 𝑦⟩ ∈ 𝑅)
17	14, 16	eqeltrd 2861	. . . . . . . . . . . . . 14 ⊢ (𝑧𝑅𝑦 → if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) ∈ 𝑅)
18		df-br 5104	. . . . . . . . . . . . . . . 16 ⊢ (𝑦𝑅𝑧 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑅)
19	18	bilani 510	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑧𝑅𝑦 ∧ 𝑦𝑅𝑧) → ⟨𝑦, 𝑧⟩ ∈ 𝑅)
20		iffalse 4491	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑧𝑅𝑦 → if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) = ⟨𝑦, 𝑧⟩)
21	20	eleq1d 2846	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑧𝑅𝑦 → (if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) ∈ 𝑅 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑅))
22	21	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑧𝑅𝑦 ∧ 𝑦𝑅𝑧) → (if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) ∈ 𝑅 ↔ ⟨𝑦, 𝑧⟩ ∈ 𝑅))
23	19, 22	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑧𝑅𝑦 ∧ 𝑦𝑅𝑧) → if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) ∈ 𝑅)
24	17, 23	jaoi3 1076	. . . . . . . . . . . . 13 ⊢ ((𝑧𝑅𝑦 ∨ 𝑦𝑅𝑧) → if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) ∈ 𝑅)
25	13, 24	syl6 36	. . . . . . . . . . . 12 ⊢ ((𝑅 Or 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ (𝐴 ∖ {𝑦}) → if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) ∈ 𝑅))
26	25	ralrimiv 3154	. . . . . . . . . . 11 ⊢ ((𝑅 Or 𝐴 ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ (𝐴 ∖ {𝑦})if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) ∈ 𝑅)
27		eldifsnneq 4754	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → ¬ 𝑤 = 𝑦)
28	27	neqcomd 2771	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → ¬ 𝑦 = 𝑤)
29		vex 3455	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
30		vex 3455	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
31	29, 30	opth1 5444	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑧, 𝑦⟩ = ⟨𝑤, 𝑦⟩ → 𝑧 = 𝑤)
32	31	a1d 26	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑧, 𝑦⟩ = ⟨𝑤, 𝑦⟩ → (¬ 𝑦 = 𝑤 → 𝑧 = 𝑤))
33	29, 30	opth 5445	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑧, 𝑦⟩ = ⟨𝑦, 𝑤⟩ ↔ (𝑧 = 𝑦 ∧ 𝑦 = 𝑤))
34	33	simprbi 503	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑧, 𝑦⟩ = ⟨𝑦, 𝑤⟩ → 𝑦 = 𝑤)
35	34	pm2.24d 152	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑧, 𝑦⟩ = ⟨𝑦, 𝑤⟩ → (¬ 𝑦 = 𝑤 → 𝑧 = 𝑤))
36	30, 29	opth1 5444	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑦⟩ → 𝑦 = 𝑤)
37	36	pm2.24d 152	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑤, 𝑦⟩ → (¬ 𝑦 = 𝑤 → 𝑧 = 𝑤))
38	30, 29	opth 5445	. . . . . . . . . . . . . . . . 17 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑤⟩ ↔ (𝑦 = 𝑦 ∧ 𝑧 = 𝑤))
39	38	simprbi 503	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑤⟩ → 𝑧 = 𝑤)
40	39	a1d 26	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑤⟩ → (¬ 𝑦 = 𝑤 → 𝑧 = 𝑤))
41	32, 35, 37, 40	jaeqifi 35710	. . . . . . . . . . . . . 14 ⊢ (if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) = if(𝑤𝑅𝑦, ⟨𝑤, 𝑦⟩, ⟨𝑦, 𝑤⟩) → (¬ 𝑦 = 𝑤 → 𝑧 = 𝑤))
42	28, 41	syl5com 32	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (𝐴 ∖ {𝑦}) → (if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) = if(𝑤𝑅𝑦, ⟨𝑤, 𝑦⟩, ⟨𝑦, 𝑤⟩) → 𝑧 = 𝑤))
43	42	rgen 3079	. . . . . . . . . . . 12 ⊢ ∀𝑤 ∈ (𝐴 ∖ {𝑦})(if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) = if(𝑤𝑅𝑦, ⟨𝑤, 𝑦⟩, ⟨𝑦, 𝑤⟩) → 𝑧 = 𝑤)
44	43	rgenw 3081	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ (𝐴 ∖ {𝑦})∀𝑤 ∈ (𝐴 ∖ {𝑦})(if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) = if(𝑤𝑅𝑦, ⟨𝑤, 𝑦⟩, ⟨𝑦, 𝑤⟩) → 𝑧 = 𝑤)
45		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝐴 ∖ {𝑦}) ↦ if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩)) = (𝑧 ∈ (𝐴 ∖ {𝑦}) ↦ if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩))
46		breq1 5106	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (𝑧𝑅𝑦 ↔ 𝑤𝑅𝑦))
47		opeq1 4833	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ⟨𝑧, 𝑦⟩ = ⟨𝑤, 𝑦⟩)
48		opeq2 4834	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → ⟨𝑦, 𝑧⟩ = ⟨𝑦, 𝑤⟩)
49	46, 47, 48	ifbieq12d 4511	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) = if(𝑤𝑅𝑦, ⟨𝑤, 𝑦⟩, ⟨𝑦, 𝑤⟩))
50	45, 49	f1mpt 7265	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (𝐴 ∖ {𝑦}) ↦ if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩)):(𝐴 ∖ {𝑦})–1-1→𝑅 ↔ (∀𝑧 ∈ (𝐴 ∖ {𝑦})if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) ∈ 𝑅 ∧ ∀𝑧 ∈ (𝐴 ∖ {𝑦})∀𝑤 ∈ (𝐴 ∖ {𝑦})(if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩) = if(𝑤𝑅𝑦, ⟨𝑤, 𝑦⟩, ⟨𝑦, 𝑤⟩) → 𝑧 = 𝑤)))
51	26, 44, 50	sylanblrc 602	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 ∈ (𝐴 ∖ {𝑦}) ↦ if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩)):(𝐴 ∖ {𝑦})–1-1→𝑅)
52		f1domg 8998	. . . . . . . . . 10 ⊢ (𝑅 ∈ 𝑉 → ((𝑧 ∈ (𝐴 ∖ {𝑦}) ↦ if(𝑧𝑅𝑦, ⟨𝑧, 𝑦⟩, ⟨𝑦, 𝑧⟩)):(𝐴 ∖ {𝑦})–1-1→𝑅 → (𝐴 ∖ {𝑦}) ≼ 𝑅))
53	51, 52	syl5 35	. . . . . . . . 9 ⊢ (𝑅 ∈ 𝑉 → ((𝑅 Or 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ≼ 𝑅))
54	53	impl 461	. . . . . . . 8 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑅 Or 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝐴 ∖ {𝑦}) ≼ 𝑅)
55		endomtr 9039	. . . . . . . 8 ⊢ ((𝐴 ≈ (𝐴 ∖ {𝑦}) ∧ (𝐴 ∖ {𝑦}) ≼ 𝑅) → 𝐴 ≼ 𝑅)
56	6, 54, 55	syl3an132 1184	. . . . . . 7 ⊢ ((ω ≼ 𝐴 ∧ (𝑅 ∈ 𝑉 ∧ 𝑅 Or 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝐴 ≼ 𝑅)
57	56	3expia 1139	. . . . . 6 ⊢ ((ω ≼ 𝐴 ∧ (𝑅 ∈ 𝑉 ∧ 𝑅 Or 𝐴)) → (𝑦 ∈ 𝐴 → 𝐴 ≼ 𝑅))
58	57	exlimdv 1966	. . . . 5 ⊢ ((ω ≼ 𝐴 ∧ (𝑅 ∈ 𝑉 ∧ 𝑅 Or 𝐴)) → (∃𝑦 𝑦 ∈ 𝐴 → 𝐴 ≼ 𝑅))
59	4, 58	mpd 16	. . . 4 ⊢ ((ω ≼ 𝐴 ∧ (𝑅 ∈ 𝑉 ∧ 𝑅 Or 𝐴)) → 𝐴 ≼ 𝑅)
60	59	ancom2s 663	. . 3 ⊢ ((ω ≼ 𝐴 ∧ (𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉)) → 𝐴 ≼ 𝑅)
61	60	ancoms 464	. 2 ⊢ (((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉) ∧ ω ≼ 𝐴) → 𝐴 ≼ 𝑅)
62	61	3impa 1127	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝑅 ∈ 𝑉 ∧ ω ≼ 𝐴) → 𝐴 ≼ 𝑅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077   ∖ cdif 3896  ∅c0 4279  ifcif 4482  {csn 4584  ⟨cop 4590   class class class wbr 5103   ↦ cmpt 5186   Or wor 5558  –1-1→wf1 6535  ωcom 7877   ≈ cen 8970   ≼ cdom 8971

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-er 8717  df-en 8974  df-dom 8975

This theorem is used by:  weexenwe  35756