Theorem ordtr2 2257

Description: Transitive law for ordinal classes.

Assertion

Ref	Expression
ordtr2	⊢ ((Ord A ∧ Ord C) → ((A ⊆ B ∧ B ∈ C) → A ∈ C))

Proof of Theorem ordtr2

Step	Hyp	Ref	Expression
1		ordsseleq 2227	. . . . . . . . . 10 ⊢ ((Ord A ∧ Ord B) → (A ⊆ B ↔ (A ∈ B ∨ A = B)))
2	1	biimpd 135	. . . . . . . . 9 ⊢ ((Ord A ∧ Ord B) → (A ⊆ B → (A ∈ B ∨ A = B)))
3		ordtr1 2256	. . . . . . . . . . . . 13 ⊢ (Ord C → ((A ∈ B ∧ B ∈ C) → A ∈ C))
4	3	exp3a 292	. . . . . . . . . . . 12 ⊢ (Ord C → (A ∈ B → (B ∈ C → A ∈ C)))
5		eleq1a 1158	. . . . . . . . . . . . . 14 ⊢ (B ∈ C → (A = B → A ∈ C))
6	5	com12 13	. . . . . . . . . . . . 13 ⊢ (A = B → (B ∈ C → A ∈ C))
7	6	a1i 7	. . . . . . . . . . . 12 ⊢ (Ord C → (A = B → (B ∈ C → A ∈ C)))
8	4, 7	jaod 329	. . . . . . . . . . 11 ⊢ (Ord C → ((A ∈ B ∨ A = B) → (B ∈ C → A ∈ C)))
9	8	com23 32	. . . . . . . . . 10 ⊢ (Ord C → (B ∈ C → ((A ∈ B ∨ A = B) → A ∈ C)))
10	9	imp 277	. . . . . . . . 9 ⊢ ((Ord C ∧ B ∈ C) → ((A ∈ B ∨ A = B) → A ∈ C))
11	2, 10	syl9 55	. . . . . . . 8 ⊢ ((Ord A ∧ Ord B) → ((Ord C ∧ B ∈ C) → (A ⊆ B → A ∈ C)))
12	11	exp 291	. . . . . . 7 ⊢ (Ord A → (Ord B → ((Ord C ∧ B ∈ C) → (A ⊆ B → A ∈ C))))
13		ordelord 2221	. . . . . . 7 ⊢ ((Ord C ∧ B ∈ C) → Ord B)
14	12, 13	syl5 22	. . . . . 6 ⊢ (Ord A → ((Ord C ∧ B ∈ C) → ((Ord C ∧ B ∈ C) → (A ⊆ B → A ∈ C))))
15	14	pm2.43d 59	. . . . 5 ⊢ (Ord A → ((Ord C ∧ B ∈ C) → (A ⊆ B → A ∈ C)))
16	15	exp3a 292	. . . 4 ⊢ (Ord A → (Ord C → (B ∈ C → (A ⊆ B → A ∈ C))))
17	16	imp 277	. . 3 ⊢ ((Ord A ∧ Ord C) → (B ∈ C → (A ⊆ B → A ∈ C)))
18	17	com23 32	. 2 ⊢ ((Ord A ∧ Ord C) → (A ⊆ B → (B ∈ C → A ∈ C)))
19	18	imp3a 279	1 ⊢ ((Ord A ∧ Ord C) → ((A ⊆ B ∧ B ∈ C) → A ∈ C))

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   ⊆ wss 1487  Ord word 2198

This theorem is referenced by:  ontr2 2259  nnarcl 3174

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-3or 582  df-3an 583  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-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202

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