Theorem wetrep 5682

Description: On a class well-ordered by membership, the membership predicate is transitive. (Contributed by NM, 22-Apr-1994.)

Assertion

Ref	Expression
wetrep	⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))

Proof of Theorem wetrep

Step	Hyp	Ref	Expression
1		weso 5680	. . 3 ⊢ ( E We 𝐴 → E Or 𝐴)
2		sotr 5622	. . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
3	1, 2	sylan 580	. 2 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
4		epel 5592	. . 3 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
5		epel 5592	. . 3 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
6	4, 5	anbi12i 628	. 2 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
7		epel 5592	. 2 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
8	3, 6, 7	3imtr3g 295	1 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2106   class class class wbr 5148   E cep 5588   Or wor 5596   We wwe 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-po 5597  df-so 5598  df-we 5643

This theorem is referenced by:  wefrc  5683  ordelord  6408