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Theorem zsupcllemstep 10589

Description: Lemma for zsupcl 10591. Induction step. (Contributed by Jim Kingdon, 7-Dec-2021.)

**Hypothesis**
Ref	Expression
zsupcllemstep.dc	⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → DECID 𝜓)

**Assertion**
Ref	Expression
zsupcllemstep	⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))

Distinct variable groups: 𝑛,𝐾,𝑥,𝑦,𝑧 𝑛,𝑀,𝑦 𝜑,𝑛,𝑦 𝜓,𝑥,𝑦,𝑧 Allowed substitution hints: 𝜑(𝑥,𝑧) 𝜓(𝑛) 𝑀(𝑥,𝑧)

**Proof of Theorem zsupcllemstep**
Step	Hyp	Ref	Expression
1		eluzelz 9863	. . . . 5 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → 𝐾 ∈ ℤ)
2	1	ad3antrrr 492	. . . 4 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) → 𝐾 ∈ ℤ)
3		nfv 1577	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐾 ∈ (ℤ_≥‘𝑀)
4		nfv 1577	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓)
5		nfcv 2384	. . . . . . . . . 10 ⊢ Ⅎ𝑦ℤ
6		nfra1 2573	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦
7		nfra1 2573	. . . . . . . . . . 11 ⊢ Ⅎ𝑦∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)
8	6, 7	nfan 1614	. . . . . . . . . 10 ⊢ Ⅎ𝑦(∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))
9	5, 8	nfrexya 2583	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))
10	4, 9	nfim 1621	. . . . . . . 8 ⊢ Ⅎ𝑦((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
11	3, 10	nfan 1614	. . . . . . 7 ⊢ Ⅎ𝑦(𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
12		nfv 1577	. . . . . . 7 ⊢ Ⅎ𝑦(𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)
13	11, 12	nfan 1614	. . . . . 6 ⊢ Ⅎ𝑦((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓))
14		nfv 1577	. . . . . 6 ⊢ Ⅎ𝑦[𝐾 / 𝑛]𝜓
15	13, 14	nfan 1614	. . . . 5 ⊢ Ⅎ𝑦(((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓)
16		nfcv 2384	. . . . . . . . . . 11 ⊢ Ⅎ𝑛ℤ
17	16	elrabsf 3081	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝑦 ∈ ℤ ∧ [𝑦 / 𝑛]𝜓))
18	17	simprbi 275	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → [𝑦 / 𝑛]𝜓)
19		sbsbc 3046	. . . . . . . . 9 ⊢ ([𝑦 / 𝑛]𝜓 ↔ [𝑦 / 𝑛]𝜓)
20	18, 19	sylibr 134	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → [𝑦 / 𝑛]𝜓)
21	20	ad2antlr 489	. . . . . . 7 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → [𝑦 / 𝑛]𝜓)
22		elrabi 2970	. . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → 𝑦 ∈ ℤ)
23		zltp1le 9632	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝐾 < 𝑦 ↔ (𝐾 + 1) ≤ 𝑦))
24	2, 22, 23	syl2an 289	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) → (𝐾 < 𝑦 ↔ (𝐾 + 1) ≤ 𝑦))
25	24	biimpa 296	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → (𝐾 + 1) ≤ 𝑦)
26	2	peano2zd 9703	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) → (𝐾 + 1) ∈ ℤ)
27		eluz 9867	. . . . . . . . . . 11 ⊢ (((𝐾 + 1) ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑦 ∈ (ℤ_≥‘(𝐾 + 1)) ↔ (𝐾 + 1) ≤ 𝑦))
28	26, 22, 27	syl2an 289	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) → (𝑦 ∈ (ℤ_≥‘(𝐾 + 1)) ↔ (𝐾 + 1) ≤ 𝑦))
29	28	adantr 276	. . . . . . . . 9 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → (𝑦 ∈ (ℤ_≥‘(𝐾 + 1)) ↔ (𝐾 + 1) ≤ 𝑦))
30	25, 29	mpbird 167	. . . . . . . 8 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → 𝑦 ∈ (ℤ_≥‘(𝐾 + 1)))
31		simprr 533	. . . . . . . . 9 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)
32	31	ad3antrrr 492	. . . . . . . 8 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)
33		nfs1v 1993	. . . . . . . . . 10 ⊢ Ⅎ𝑛[𝑦 / 𝑛]𝜓
34	33	nfn 1706	. . . . . . . . 9 ⊢ Ⅎ𝑛 ¬ [𝑦 / 𝑛]𝜓
35		sbequ12 1820	. . . . . . . . . 10 ⊢ (𝑛 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑛]𝜓))
36	35	notbid 673	. . . . . . . . 9 ⊢ (𝑛 = 𝑦 → (¬ 𝜓 ↔ ¬ [𝑦 / 𝑛]𝜓))
37	34, 36	rspc 2915	. . . . . . . 8 ⊢ (𝑦 ∈ (ℤ_≥‘(𝐾 + 1)) → (∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓 → ¬ [𝑦 / 𝑛]𝜓))
38	30, 32, 37	sylc 62	. . . . . . 7 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) ∧ 𝐾 < 𝑦) → ¬ [𝑦 / 𝑛]𝜓)
39	21, 38	pm2.65da 667	. . . . . 6 ⊢ (((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓}) → ¬ 𝐾 < 𝑦)
40	39	ex 115	. . . . 5 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) → (𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} → ¬ 𝐾 < 𝑦))
41	15, 40	ralrimi 2613	. . . 4 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) → ∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦)
42	2	ad2antrr 488	. . . . . . . 8 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 𝐾) → 𝐾 ∈ ℤ)
43		simpllr 536	. . . . . . . 8 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 𝐾) → [𝐾 / 𝑛]𝜓)
44	16	elrabsf 3081	. . . . . . . 8 ⊢ (𝐾 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ↔ (𝐾 ∈ ℤ ∧ [𝐾 / 𝑛]𝜓))
45	42, 43, 44	sylanbrc 417	. . . . . . 7 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 𝐾) → 𝐾 ∈ {𝑛 ∈ ℤ ∣ 𝜓})
46		breq2 4113	. . . . . . . 8 ⊢ (𝑧 = 𝐾 → (𝑦 < 𝑧 ↔ 𝑦 < 𝐾))
47	46	rspcev 2921	. . . . . . 7 ⊢ ((𝐾 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ∧ 𝑦 < 𝐾) → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)
48	45, 47	sylancom 420	. . . . . 6 ⊢ ((((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < 𝐾) → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)
49	48	exp31 364	. . . . 5 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) → (𝑦 ∈ ℝ → (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
50	15, 49	ralrimi 2613	. . . 4 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) → ∀𝑦 ∈ ℝ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))
51		breq1 4112	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑥 < 𝑦 ↔ 𝐾 < 𝑦))
52	51	notbid 673	. . . . . . 7 ⊢ (𝑥 = 𝐾 → (¬ 𝑥 < 𝑦 ↔ ¬ 𝐾 < 𝑦))
53	52	ralbidv 2542	. . . . . 6 ⊢ (𝑥 = 𝐾 → (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦))
54		breq2 4113	. . . . . . . 8 ⊢ (𝑥 = 𝐾 → (𝑦 < 𝑥 ↔ 𝑦 < 𝐾))
55	54	imbi1d 231	. . . . . . 7 ⊢ (𝑥 = 𝐾 → ((𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧) ↔ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
56	55	ralbidv 2542	. . . . . 6 ⊢ (𝑥 = 𝐾 → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
57	53, 56	anbi12d 473	. . . . 5 ⊢ (𝑥 = 𝐾 → ((∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)) ↔ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
58	57	rspcev 2921	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝐾 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝐾 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
59	2, 41, 50, 58	syl12anc 1272	. . 3 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ [𝐾 / 𝑛]𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
60		sbcng 3083	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ¬ [𝐾 / 𝑛]𝜓))
61	60	ad2antrr 488	. . . . . . 7 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ¬ [𝐾 / 𝑛]𝜓))
62	61	biimpar 297	. . . . . 6 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ ¬ [𝐾 / 𝑛]𝜓) → [𝐾 / 𝑛] ¬ 𝜓)
63		sbcsng 3748	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ∀𝑛 ∈ {𝐾} ¬ 𝜓))
64	63	ad3antrrr 492	. . . . . 6 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ([𝐾 / 𝑛] ¬ 𝜓 ↔ ∀𝑛 ∈ {𝐾} ¬ 𝜓))
65	62, 64	mpbid 147	. . . . 5 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ∀𝑛 ∈ {𝐾} ¬ 𝜓)
66		simplrr 538	. . . . 5 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)
67		uzid 9868	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → 𝐾 ∈ (ℤ_≥‘𝐾))
68		peano2uz 9915	. . . . . . . . . . 11 ⊢ (𝐾 ∈ (ℤ_≥‘𝐾) → (𝐾 + 1) ∈ (ℤ_≥‘𝐾))
69	67, 68	syl 14	. . . . . . . . . 10 ⊢ (𝐾 ∈ ℤ → (𝐾 + 1) ∈ (ℤ_≥‘𝐾))
70		fzouzsplit 10515	. . . . . . . . . 10 ⊢ ((𝐾 + 1) ∈ (ℤ_≥‘𝐾) → (ℤ_≥‘𝐾) = ((𝐾..^(𝐾 + 1)) ∪ (ℤ_≥‘(𝐾 + 1))))
71	1, 69, 70	3syl 17	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝐾) = ((𝐾..^(𝐾 + 1)) ∪ (ℤ_≥‘(𝐾 + 1))))
72		fzosn 10550	. . . . . . . . . . 11 ⊢ (𝐾 ∈ ℤ → (𝐾..^(𝐾 + 1)) = {𝐾})
73	1, 72	syl 14	. . . . . . . . . 10 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝐾..^(𝐾 + 1)) = {𝐾})
74	73	uneq1d 3372	. . . . . . . . 9 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → ((𝐾..^(𝐾 + 1)) ∪ (ℤ_≥‘(𝐾 + 1))) = ({𝐾} ∪ (ℤ_≥‘(𝐾 + 1))))
75	71, 74	eqtrd 2265	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝐾) = ({𝐾} ∪ (ℤ_≥‘(𝐾 + 1))))
76	75	raleqdv 2747	. . . . . . 7 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓 ↔ ∀𝑛 ∈ ({𝐾} ∪ (ℤ_≥‘(𝐾 + 1))) ¬ 𝜓))
77		ralunb 3400	. . . . . . 7 ⊢ (∀𝑛 ∈ ({𝐾} ∪ (ℤ_≥‘(𝐾 + 1))) ¬ 𝜓 ↔ (∀𝑛 ∈ {𝐾} ¬ 𝜓 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓))
78	76, 77	bitrdi 196	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓 ↔ (∀𝑛 ∈ {𝐾} ¬ 𝜓 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)))
79	78	ad3antrrr 492	. . . . 5 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ ¬ [𝐾 / 𝑛]𝜓) → (∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓 ↔ (∀𝑛 ∈ {𝐾} ¬ 𝜓 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)))
80	65, 66, 79	mpbir2and 953	. . . 4 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓)
81		simprl 531	. . . . . 6 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → 𝜑)
82		simplr 529	. . . . . 6 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
83	81, 82	mpand 429	. . . . 5 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → (∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
84	83	adantr 276	. . . 4 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ ¬ [𝐾 / 𝑛]𝜓) → (∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓 → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))))
85	80, 84	mpd 13	. . 3 ⊢ ((((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) ∧ ¬ [𝐾 / 𝑛]𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
86		zsupcllemstep.dc	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑀)) → DECID 𝜓)
87	86	ralrimiva 2615	. . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ (ℤ_≥‘𝑀)DECID 𝜓)
88	81, 87	syl 14	. . . . 5 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → ∀𝑛 ∈ (ℤ_≥‘𝑀)DECID 𝜓)
89		nfsbc1v 3061	. . . . . . . 8 ⊢ Ⅎ𝑛[𝐾 / 𝑛]𝜓
90	89	nfdc 1707	. . . . . . 7 ⊢ Ⅎ𝑛DECID [𝐾 / 𝑛]𝜓
91		sbceq1a 3052	. . . . . . . 8 ⊢ (𝑛 = 𝐾 → (𝜓 ↔ [𝐾 / 𝑛]𝜓))
92	91	dcbid 846	. . . . . . 7 ⊢ (𝑛 = 𝐾 → (DECID 𝜓 ↔ DECID [𝐾 / 𝑛]𝜓))
93	90, 92	rspc 2915	. . . . . 6 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (∀𝑛 ∈ (ℤ_≥‘𝑀)DECID 𝜓 → DECID [𝐾 / 𝑛]𝜓))
94	93	ad2antrr 488	. . . . 5 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → (∀𝑛 ∈ (ℤ_≥‘𝑀)DECID 𝜓 → DECID [𝐾 / 𝑛]𝜓))
95	88, 94	mpd 13	. . . 4 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → DECID [𝐾 / 𝑛]𝜓)
96		exmiddc 844	. . . 4 ⊢ (DECID [𝐾 / 𝑛]𝜓 → ([𝐾 / 𝑛]𝜓 ∨ ¬ [𝐾 / 𝑛]𝜓))
97	95, 96	syl 14	. . 3 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → ([𝐾 / 𝑛]𝜓 ∨ ¬ [𝐾 / 𝑛]𝜓))
98	59, 85, 97	mpjaodan 806	. 2 ⊢ (((𝐾 ∈ (ℤ_≥‘𝑀) ∧ ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))) ∧ (𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓)) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))
99	98	exp31 364	1 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘𝐾) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧))) → ((𝜑 ∧ ∀𝑛 ∈ (ℤ_≥‘(𝐾 + 1)) ¬ 𝜓) → ∃𝑥 ∈ ℤ (∀𝑦 ∈ {𝑛 ∈ ℤ ∣ 𝜓} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑛 ∈ ℤ ∣ 𝜓}𝑦 < 𝑧)))))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 = wceq 1398 [wsb 1811 ∈ wcel 2203 ∀wral 2520 ∃wrex 2521 {crab 2524 [wsbc 3042 ∪ cun 3209 {csn 3689 class class class wbr 4109 ‘cfv 5352 (class class class)co 6050 ℝcr 8126 1c1 8128 + caddc 8130 < clt 8308 ≤ cle 8309 ℤcz 9577 ℤ_≥cuz 9853 ..^cfzo 10476

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 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-addcom 8227 ax-addass 8229 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-0id 8235 ax-rnegex 8236 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243

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 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-inn 9238 df-n0 9497 df-z 9578 df-uz 9854 df-fz 10343 df-fzo 10477

This theorem is referenced by: zsupcllemex 10590

W3C validator