Theorem sucneqond 37989

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 6446	. . . . 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 2866	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
6		onsuc 7810	. . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On)
7	1, 6	syl 18	. . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On)
8	4, 7	eqeltrd 2863	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
9		eloni 6372	. . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋)
10	8, 9	syl 18	. . . . 5 ⊢ (𝜑 → Ord 𝑋)
11		ordirr 6380	. . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋)
12	10, 11	syl 18	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋)
13		eleq1 2851	. . . . . 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 2965	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Ord word 6361  Oncon0 6362  suc csuc 6364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366  df-suc 6368

This theorem is referenced by:  sucneqoni  37990