Theorem ordnbtwn 2316

Description: There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41.

Assertion

Ref	Expression
ordnbtwn	⊢ (Ord A → ¬ (A ∈ B ∧ B ∈ suc A))

Proof of Theorem ordnbtwn

Step	Hyp	Ref	Expression
1		ordn2lp 2219	. . . 4 ⊢ (Ord A → ¬ (A ∈ B ∧ B ∈ A))
2		ordeirr 2217	. . . 4 ⊢ (Ord A → ¬ A ∈ A)
3	1, 2	jca 236	. . 3 ⊢ (Ord A → (¬ (A ∈ B ∧ B ∈ A) ∧ ¬ A ∈ A))
4		ioran 254	. . 3 ⊢ (¬ ((A ∈ B ∧ B ∈ A) ∨ A ∈ A) ↔ (¬ (A ∈ B ∧ B ∈ A) ∧ ¬ A ∈ A))
5	3, 4	sylibr 175	. 2 ⊢ (Ord A → ¬ ((A ∈ B ∧ B ∈ A) ∨ A ∈ A))
6		elsuci 2289	. . . . 5 ⊢ (B ∈ suc A → (B ∈ A ∨ B = A))
7	6	anim2i 270	. . . 4 ⊢ ((A ∈ B ∧ B ∈ suc A) → (A ∈ B ∧ (B ∈ A ∨ B = A)))
8		andi 456	. . . 4 ⊢ ((A ∈ B ∧ (B ∈ A ∨ B = A)) ↔ ((A ∈ B ∧ B ∈ A) ∨ (A ∈ B ∧ B = A)))
9	7, 8	sylib 173	. . 3 ⊢ ((A ∈ B ∧ B ∈ suc A) → ((A ∈ B ∧ B ∈ A) ∨ (A ∈ B ∧ B = A)))
10		eleq2 1150	. . . . 5 ⊢ (B = A → (A ∈ B ↔ A ∈ A))
11	10	biimpac 326	. . . 4 ⊢ ((A ∈ B ∧ B = A) → A ∈ A)
12	11	orim2i 273	. . 3 ⊢ (((A ∈ B ∧ B ∈ A) ∨ (A ∈ B ∧ B = A)) → ((A ∈ B ∧ B ∈ A) ∨ A ∈ A))
13	9, 12	syl 12	. 2 ⊢ ((A ∈ B ∧ B ∈ suc A) → ((A ∈ B ∧ B ∈ A) ∨ A ∈ A))
14	5, 13	nsyl 102	1 ⊢ (Ord A → ¬ (A ∈ B ∧ B ∈ suc A))

Syntax hints:  ¬ wn 1   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Ord word 2198  suc csuc 2201

This theorem is referenced by:  onnbtwn 2317  ordsucss 2320

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169  df-we 2186  df-ord 2202  df-suc 2205

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