sucneqond - Mathbox for ML

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Theorem sucneqond 37823

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 6425	. . . . 5 ⊢ (𝑌 ∈ On → 𝑌 ∈ suc 𝑌)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝑌 ∈ suc 𝑌)
4		sucneqond.1	. . . 4 ⊢ (𝜑 → 𝑋 = suc 𝑌)
5	3, 4	eleqtrrd 2864	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
6		onsuc 7789	. . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On)
7	1, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On)
8	4, 7	eqeltrd 2861	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
9		eloni 6352	. . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → Ord 𝑋)
11		ordirr 6360	. . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋)
13		eleq1 2849	. . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑋 ↔ 𝑌 ∈ 𝑋))
14	13	biimprd 250	. . . . 5 ⊢ (𝑋 = 𝑌 → (𝑌 ∈ 𝑋 → 𝑋 ∈ 𝑋))
15	14	con3d 152	. . . 4 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 ∈ 𝑋 → ¬ 𝑌 ∈ 𝑋))
16	12, 15	syl5com 31	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 → ¬ 𝑌 ∈ 𝑋))
17	5, 16	mt2d 136	. 2 ⊢ (𝜑 → ¬ 𝑋 = 𝑌)
18	17	neqned 2963	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 Ord word 6341 Oncon0 6342 suc csuc 6344

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-ext 2733 ax-sep 5245 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-sb 2090 df-clab 2740 df-cleq 2753 df-clel 2836 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-tr 5207 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-ord 6345 df-on 6346 df-suc 6348

This theorem is referenced by: sucneqoni 37824