Theorem sucneqond 37398

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 6389	. . . . 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 2834	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
6		onsuc 7743	. . . . . . . 8 ⊢ (𝑌 ∈ On → suc 𝑌 ∈ On)
7	1, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → suc 𝑌 ∈ On)
8	4, 7	eqeltrd 2831	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ On)
9		eloni 6316	. . . . . 6 ⊢ (𝑋 ∈ On → Ord 𝑋)
10	8, 9	syl 17	. . . . 5 ⊢ (𝜑 → Ord 𝑋)
11		ordirr 6324	. . . . 5 ⊢ (Ord 𝑋 → ¬ 𝑋 ∈ 𝑋)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑋)
13		eleq1 2819	. . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 ∈ 𝑋 ↔ 𝑌 ∈ 𝑋))
14	13	biimprd 248	. . . . 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 2935	1 ⊢ (𝜑 → 𝑋 ≠ 𝑌)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  Ord word 6305  Oncon0 6306  suc csuc 6308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-ord 6309  df-on 6310  df-suc 6312

This theorem is referenced by:  sucneqoni  37399