sucneqond - Mathbox for ML

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Theorem sucneqond 38127

Description: Inequality of an ordinal set with its successor. Does not use the axiom of regularity. (Contributed by ML, 18-Oct-2020.)

**Hypotheses**
Ref	Expression
sucneqond.1	⊢ (𝜑 → 𝑋 = suc 𝑌)
sucneqond.2	⊢ (𝜑 → 𝑌 ∈ On)

**Assertion**
Ref	Expression
sucneqond	⊢ (𝜑 → 𝑋 ≠ 𝑌)

**Proof of Theorem sucneqond**
Step	Hyp	Ref	Expression
1		sucneqond.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ On)
2		sucidg 6445	. . . . 5 ⊢ (𝑌 ∈ On → 𝑌 ∈ suc 𝑌)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → 𝑌 ∈ suc 𝑌)
4		sucneqond.1	. . . 4 ⊢ (𝜑 → 𝑋 = suc 𝑌)
5	3, 4	eleqtrrd 2865	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
6		onsuc 7813	. . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On)
7	1, 6	syl 18	. . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On)
8	4, 7	eqeltrd 2862	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
9		eloni 6371	. . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋)
10	8, 9	syl 18	. . . . 5 ⊢ (𝜑 → Ord 𝑋)
11		ordirr 6379	. . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋)
12	10, 11	syl 18	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋)
13		eleq1 2850	. . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑋 ↔ 𝑌 ∈ 𝑋))
14	13	biimprd 251	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑌 ∈ 𝑋 → 𝑋 ∈ 𝑋))
15	14	con3d 153	. . . 4 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 ∈ 𝑋 → ¬ 𝑌 ∈ 𝑋))
16	12, 15	syl5com 32	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌 ∈ 𝑋))
17	5, 16	mt2d 137	. 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌)
18	17	neqned 2964	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 Ord word 6360 Oncon0 6361 suc csuc 6363

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-ext 2734 ax-sep 5255 ax-pr 5402 ax-un 7740

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-sb 2100 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-opab 5172 df-tr 5217 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-ord 6364 df-on 6365 df-suc 6367

This theorem is used by: sucneqoni 38128