exmidsbthrlem - Mathbox for Jim Kingdon

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Theorem exmidsbthrlem 16730

Description: Lemma for exmidsbthr 16731. (Contributed by Jim Kingdon, 11-Aug-2022.)

**Hypothesis**
Ref	Expression
exmidsbthrlem.s	⊢ 𝑆 = (𝑝 ∈ ℕ_∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1_o, (𝑝‘∪ 𝑖))))

**Assertion**
Ref	Expression
exmidsbthrlem	⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID)

Distinct variable groups: 𝑆,𝑖 𝑖,𝑝 𝑥,𝑦 Allowed substitution hints: 𝑆(𝑥,𝑦,𝑝)

Proof of Theorem exmidsbthrlem Dummy variables 𝑎 𝑏 𝑘 𝑧 𝑓 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 109	. . . . . . 7 ⊢ ((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) → ∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦))
2		nninfex 7363	. . . . . . . . . 10 ⊢ ℕ_∞ ∈ V
3		fconstmpt 4779	. . . . . . . . . . . . . . 15 ⊢ (ω × {∅}) = (𝑖 ∈ ω ↦ ∅)
4		0nninf 16710	. . . . . . . . . . . . . . 15 ⊢ (ω × {∅}) ∈ ℕ_∞
5	3, 4	eqeltrri 2305	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ ω ↦ ∅) ∈ ℕ_∞
6	5	fconst6 5545	. . . . . . . . . . . . 13 ⊢ (𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧⟶ℕ_∞
7	6	a1i 9	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ {∅} → (𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧⟶ℕ_∞)
8		ssel 3222	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ⊆ {∅} → (𝑢 ∈ 𝑧 → 𝑢 ∈ {∅}))
9		elsni 3691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ {∅} → 𝑢 = ∅)
10	8, 9	syl6 33	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ⊆ {∅} → (𝑢 ∈ 𝑧 → 𝑢 = ∅))
11		ssel 3222	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ⊆ {∅} → (𝑣 ∈ 𝑧 → 𝑣 ∈ {∅}))
12		elsni 3691	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑣 ∈ {∅} → 𝑣 = ∅)
13	11, 12	syl6 33	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ⊆ {∅} → (𝑣 ∈ 𝑧 → 𝑣 = ∅))
14	10, 13	anim12d 335	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ⊆ {∅} → ((𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧) → (𝑢 = ∅ ∧ 𝑣 = ∅)))
15		eqtr3 2251	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 = ∅ ∧ 𝑣 = ∅) → 𝑢 = 𝑣)
16	14, 15	syl6 33	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ⊆ {∅} → ((𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧) → 𝑢 = 𝑣))
17	16	imp 124	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ⊆ {∅} ∧ (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧)) → 𝑢 = 𝑣)
18	17	a1d 22	. . . . . . . . . . . . 13 ⊢ ((𝑧 ⊆ {∅} ∧ (𝑢 ∈ 𝑧 ∧ 𝑣 ∈ 𝑧)) → (((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑢) = ((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑣) → 𝑢 = 𝑣))
19	18	ralrimivva 2615	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ {∅} → ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑢) = ((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑣) → 𝑢 = 𝑣))
20		dff13 5919	. . . . . . . . . . . 12 ⊢ ((𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧–1-1→ℕ_∞ ↔ ((𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧⟶ℕ_∞ ∧ ∀𝑢 ∈ 𝑧 ∀𝑣 ∈ 𝑧 (((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑢) = ((𝑧 × {(𝑖 ∈ ω ↦ ∅)})‘𝑣) → 𝑢 = 𝑣)))
21	7, 19, 20	sylanbrc 417	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ {∅} → (𝑧 × {(𝑖 ∈ ω ↦ ∅)}):𝑧–1-1→ℕ_∞)
22		exmidsbthrlem.s	. . . . . . . . . . . . 13 ⊢ 𝑆 = (𝑝 ∈ ℕ_∞ ↦ (𝑖 ∈ ω ↦ if(𝑖 = ∅, 1_o, (𝑝‘∪ 𝑖))))
23	22	peano4nninf 16712	. . . . . . . . . . . 12 ⊢ 𝑆:ℕ_∞–1-1→ℕ_∞
24	23	a1i 9	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ {∅} → 𝑆:ℕ_∞–1-1→ℕ_∞)
25		disj 3545	. . . . . . . . . . . . 13 ⊢ ((ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ∩ ran 𝑆) = ∅ ↔ ∀𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ¬ 𝑎 ∈ ran 𝑆)
26	22	peano3nninf 16713	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑏 ∈ ℕ_∞ → (𝑆‘𝑏) ≠ (𝑘 ∈ ω ↦ ∅))
27		eqidd 2232	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → ∅ = ∅)
28	27	cbvmptv 4190	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ ω ↦ ∅) = (𝑖 ∈ ω ↦ ∅)
29	28	neeq2i 2419	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆‘𝑏) ≠ (𝑘 ∈ ω ↦ ∅) ↔ (𝑆‘𝑏) ≠ (𝑖 ∈ ω ↦ ∅))
30	26, 29	sylib 122	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ℕ_∞ → (𝑆‘𝑏) ≠ (𝑖 ∈ ω ↦ ∅))
31	30	neneqd 2424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ ℕ_∞ → ¬ (𝑆‘𝑏) = (𝑖 ∈ ω ↦ ∅))
32	31	nrex 2625	. . . . . . . . . . . . . . . 16 ⊢ ¬ ∃𝑏 ∈ ℕ_∞ (𝑆‘𝑏) = (𝑖 ∈ ω ↦ ∅)
33		f1dm 5556	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆:ℕ_∞–1-1→ℕ_∞ → dom 𝑆 = ℕ_∞)
34	23, 33	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ dom 𝑆 = ℕ_∞
35		eqcom 2233	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ ω ↦ ∅) = (𝑆‘𝑏) ↔ (𝑆‘𝑏) = (𝑖 ∈ ω ↦ ∅))
36	34, 35	rexeqbii 2546	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑏 ∈ dom 𝑆(𝑖 ∈ ω ↦ ∅) = (𝑆‘𝑏) ↔ ∃𝑏 ∈ ℕ_∞ (𝑆‘𝑏) = (𝑖 ∈ ω ↦ ∅))
37	32, 36	mtbir 678	. . . . . . . . . . . . . . 15 ⊢ ¬ ∃𝑏 ∈ dom 𝑆(𝑖 ∈ ω ↦ ∅) = (𝑆‘𝑏)
38	22	funmpt2 5372	. . . . . . . . . . . . . . . 16 ⊢ Fun 𝑆
39		elrnrexdm 5794	. . . . . . . . . . . . . . . 16 ⊢ (Fun 𝑆 → ((𝑖 ∈ ω ↦ ∅) ∈ ran 𝑆 → ∃𝑏 ∈ dom 𝑆(𝑖 ∈ ω ↦ ∅) = (𝑆‘𝑏)))
40	38, 39	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ ω ↦ ∅) ∈ ran 𝑆 → ∃𝑏 ∈ dom 𝑆(𝑖 ∈ ω ↦ ∅) = (𝑆‘𝑏))
41	37, 40	mto 668	. . . . . . . . . . . . . 14 ⊢ ¬ (𝑖 ∈ ω ↦ ∅) ∈ ran 𝑆
42		rnxpss 5175	. . . . . . . . . . . . . . . . 17 ⊢ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ⊆ {(𝑖 ∈ ω ↦ ∅)}
43	42	sseli 3224	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) → 𝑎 ∈ {(𝑖 ∈ ω ↦ ∅)})
44		elsni 3691	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 ∈ {(𝑖 ∈ ω ↦ ∅)} → 𝑎 = (𝑖 ∈ ω ↦ ∅))
45	43, 44	syl 14	. . . . . . . . . . . . . . 15 ⊢ (𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) → 𝑎 = (𝑖 ∈ ω ↦ ∅))
46	45	eleq1d 2300	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) → (𝑎 ∈ ran 𝑆 ↔ (𝑖 ∈ ω ↦ ∅) ∈ ran 𝑆))
47	41, 46	mtbiri 682	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) → ¬ 𝑎 ∈ ran 𝑆)
48	25, 47	mprgbir 2591	. . . . . . . . . . . 12 ⊢ (ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ∩ ran 𝑆) = ∅
49	48	a1i 9	. . . . . . . . . . 11 ⊢ (𝑧 ⊆ {∅} → (ran (𝑧 × {(𝑖 ∈ ω ↦ ∅)}) ∩ ran 𝑆) = ∅)
50	21, 24, 49	casef1 7332	. . . . . . . . . 10 ⊢ (𝑧 ⊆ {∅} → case((𝑧 × {(𝑖 ∈ ω ↦ ∅)}), 𝑆):(𝑧 ⊔ ℕ_∞)–1-1→ℕ_∞)
51		f1domg 6974	. . . . . . . . . 10 ⊢ (ℕ_∞ ∈ V → (case((𝑧 × {(𝑖 ∈ ω ↦ ∅)}), 𝑆):(𝑧 ⊔ ℕ_∞)–1-1→ℕ_∞ → (𝑧 ⊔ ℕ_∞) ≼ ℕ_∞))
52	2, 50, 51	mpsyl 65	. . . . . . . . 9 ⊢ (𝑧 ⊆ {∅} → (𝑧 ⊔ ℕ_∞) ≼ ℕ_∞)
53	52	adantl 277	. . . . . . . 8 ⊢ ((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) → (𝑧 ⊔ ℕ_∞) ≼ ℕ_∞)
54		inrresf1 7304	. . . . . . . . 9 ⊢ (inr ↾ ℕ_∞):ℕ_∞–1-1→(𝑧 ⊔ ℕ_∞)
55		vex 2806	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
56		djuex 7285	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ ℕ_∞ ∈ V) → (𝑧 ⊔ ℕ_∞) ∈ V)
57	55, 2, 56	mp2an 426	. . . . . . . . . 10 ⊢ (𝑧 ⊔ ℕ_∞) ∈ V
58	57	f1dom 6976	. . . . . . . . 9 ⊢ ((inr ↾ ℕ_∞):ℕ_∞–1-1→(𝑧 ⊔ ℕ_∞) → ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞))
59	54, 58	ax-mp 5	. . . . . . . 8 ⊢ ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞)
60	53, 59	jctir 313	. . . . . . 7 ⊢ ((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) → ((𝑧 ⊔ ℕ_∞) ≼ ℕ_∞ ∧ ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞)))
61		breq12 4098	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑧 ⊔ ℕ_∞) ∧ 𝑦 = ℕ_∞) → (𝑥 ≼ 𝑦 ↔ (𝑧 ⊔ ℕ_∞) ≼ ℕ_∞))
62		breq12 4098	. . . . . . . . . . . 12 ⊢ ((𝑦 = ℕ_∞ ∧ 𝑥 = (𝑧 ⊔ ℕ_∞)) → (𝑦 ≼ 𝑥 ↔ ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞)))
63	62	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑧 ⊔ ℕ_∞) ∧ 𝑦 = ℕ_∞) → (𝑦 ≼ 𝑥 ↔ ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞)))
64	61, 63	anbi12d 473	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑧 ⊔ ℕ_∞) ∧ 𝑦 = ℕ_∞) → ((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) ↔ ((𝑧 ⊔ ℕ_∞) ≼ ℕ_∞ ∧ ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞))))
65		breq12 4098	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑧 ⊔ ℕ_∞) ∧ 𝑦 = ℕ_∞) → (𝑥 ≈ 𝑦 ↔ (𝑧 ⊔ ℕ_∞) ≈ ℕ_∞))
66	64, 65	imbi12d 234	. . . . . . . . 9 ⊢ ((𝑥 = (𝑧 ⊔ ℕ_∞) ∧ 𝑦 = ℕ_∞) → (((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ↔ (((𝑧 ⊔ ℕ_∞) ≼ ℕ_∞ ∧ ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞)) → (𝑧 ⊔ ℕ_∞) ≈ ℕ_∞)))
67	66	spc2gv 2898	. . . . . . . 8 ⊢ (((𝑧 ⊔ ℕ_∞) ∈ V ∧ ℕ_∞ ∈ V) → (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → (((𝑧 ⊔ ℕ_∞) ≼ ℕ_∞ ∧ ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞)) → (𝑧 ⊔ ℕ_∞) ≈ ℕ_∞)))
68	57, 2, 67	mp2an 426	. . . . . . 7 ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → (((𝑧 ⊔ ℕ_∞) ≼ ℕ_∞ ∧ ℕ_∞ ≼ (𝑧 ⊔ ℕ_∞)) → (𝑧 ⊔ ℕ_∞) ≈ ℕ_∞))
69	1, 60, 68	sylc 62	. . . . . 6 ⊢ ((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) → (𝑧 ⊔ ℕ_∞) ≈ ℕ_∞)
70		bren 6960	. . . . . 6 ⊢ ((𝑧 ⊔ ℕ_∞) ≈ ℕ_∞ ↔ ∃𝑓 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞)
71	69, 70	sylib 122	. . . . 5 ⊢ ((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) → ∃𝑓 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞)
72		nninfomni 16725	. . . . . . . . 9 ⊢ ℕ_∞ ∈ Omni
73	72	a1i 9	. . . . . . . 8 ⊢ (((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞) → ℕ_∞ ∈ Omni)
74		f1ocnv 5605	. . . . . . . . . 10 ⊢ (𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞ → ^◡𝑓:ℕ_∞–1-1-onto→(𝑧 ⊔ ℕ_∞))
75		f1ofo 5599	. . . . . . . . . 10 ⊢ (^◡𝑓:ℕ_∞–1-1-onto→(𝑧 ⊔ ℕ_∞) → ^◡𝑓:ℕ_∞–onto→(𝑧 ⊔ ℕ_∞))
76	74, 75	syl 14	. . . . . . . . 9 ⊢ (𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞ → ^◡𝑓:ℕ_∞–onto→(𝑧 ⊔ ℕ_∞))
77	76	adantl 277	. . . . . . . 8 ⊢ (((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞) → ^◡𝑓:ℕ_∞–onto→(𝑧 ⊔ ℕ_∞))
78	73, 77	fodjuomni 7391	. . . . . . 7 ⊢ (((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞) → (∃𝑤 𝑤 ∈ 𝑧 ∨ 𝑧 = ∅))
79		sssnm 3842	. . . . . . . . . 10 ⊢ (∃𝑤 𝑤 ∈ 𝑧 → (𝑧 ⊆ {∅} ↔ 𝑧 = {∅}))
80	79	biimpcd 159	. . . . . . . . 9 ⊢ (𝑧 ⊆ {∅} → (∃𝑤 𝑤 ∈ 𝑧 → 𝑧 = {∅}))
81	80	ad2antlr 489	. . . . . . . 8 ⊢ (((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞) → (∃𝑤 𝑤 ∈ 𝑧 → 𝑧 = {∅}))
82	81	orim1d 795	. . . . . . 7 ⊢ (((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞) → ((∃𝑤 𝑤 ∈ 𝑧 ∨ 𝑧 = ∅) → (𝑧 = {∅} ∨ 𝑧 = ∅)))
83	78, 82	mpd 13	. . . . . 6 ⊢ (((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞) → (𝑧 = {∅} ∨ 𝑧 = ∅))
84	83	orcomd 737	. . . . 5 ⊢ (((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) ∧ 𝑓:(𝑧 ⊔ ℕ_∞)–1-1-onto→ℕ_∞) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
85	71, 84	exlimddv 1947	. . . 4 ⊢ ((∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) ∧ 𝑧 ⊆ {∅}) → (𝑧 = ∅ ∨ 𝑧 = {∅}))
86	85	ex 115	. . 3 ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → (𝑧 ⊆ {∅} → (𝑧 = ∅ ∨ 𝑧 = {∅})))
87	86	alrimiv 1922	. 2 ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → ∀𝑧(𝑧 ⊆ {∅} → (𝑧 = ∅ ∨ 𝑧 = {∅})))
88		exmid01 4294	. 2 ⊢ (EXMID ↔ ∀𝑧(𝑧 ⊆ {∅} → (𝑧 = ∅ ∨ 𝑧 = {∅})))
89	87, 88	sylibr 134	1 ⊢ (∀𝑥∀𝑦((𝑥 ≼ 𝑦 ∧ 𝑦 ≼ 𝑥) → 𝑥 ≈ 𝑦) → EXMID)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 ∀wal 1396 = wceq 1398 ∃wex 1541 ∈ wcel 2202 ≠ wne 2403 ∀wral 2511 ∃wrex 2512 Vcvv 2803 ∩ cin 3200 ⊆ wss 3201 ∅c0 3496 ifcif 3607 {csn 3673 ∪ cuni 3898 class class class wbr 4093 ↦ cmpt 4155 EXMIDwem 4290 ωcom 4694 × cxp 4729 ^◡ccnv 4730 dom cdm 4731 ran crn 4732 ↾ cres 4733 Fun wfun 5327 ⟶wf 5329 –1-1→wf1 5330 –onto→wfo 5331 –1-1-onto→wf1o 5332 ‘cfv 5333 1_oc1o 6618 ≈ cen 6950 ≼ cdom 6951 ⊔ cdju 7279 inrcinr 7288 casecdjucase 7325 ℕ_∞xnninf 7361 Omnicomni 7376

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-exmid 4291 df-id 4396 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-1o 6625 df-2o 6626 df-map 6862 df-en 6953 df-dom 6954 df-dju 7280 df-inl 7289 df-inr 7290 df-case 7326 df-nninf 7362 df-omni 7377

This theorem is referenced by: exmidsbthr 16731

W3C validator