Theorem limnsuc 43708

Description: A limit ordinal is not an element of the class of successor ordinals. Definition 1.11 of [Schloeder] p. 2. (Contributed by RP, 16-Jan-2025.)

Assertion

Ref	Expression
limnsuc	⊢ (Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

Distinct variable group:   𝐴,𝑎,𝑏

Proof of Theorem limnsuc

Step	Hyp	Ref	Expression
1		dflim6 43707	. 2 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
2		simp3 1139	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) → ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏)
3		eqeq1 2741	. . . . . 6 ⊢ (𝑎 = 𝐴 → (𝑎 = suc 𝑏 ↔ 𝐴 = suc 𝑏))
4	3	rexbidv 3162	. . . . 5 ⊢ (𝑎 = 𝐴 → (∃𝑏 ∈ On 𝑎 = suc 𝑏 ↔ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
5	4	elrab 3635	. . . 4 ⊢ (𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (𝐴 ∈ On ∧ ∃𝑏 ∈ On 𝐴 = suc 𝑏))
6	5	simprbi 497	. . 3 ⊢ (𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} → ∃𝑏 ∈ On 𝐴 = suc 𝑏)
7	2, 6	nsyl 140	. 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ≠ ∅ ∧ ¬ ∃𝑏 ∈ On 𝐴 = suc 𝑏) → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})
8	1, 7	sylbi 217	1 ⊢ (Lim 𝐴 → ¬ 𝐴 ∈ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏})

Syntax hints:  ¬ wn 3   → wi 4   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  {crab 3390  ∅c0 4274  Ord word 6314  Oncon0 6315  Lim wlim 6316  suc csuc 6317

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321

This theorem is referenced by: (None)