Theorem wetrep 5512

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 5510	. . 3 ⊢ ( E We 𝐴 → E Or 𝐴)
2		sotr 5461	. . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
3	1, 2	sylan 583	. 2 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
4		epel 5433	. . 3 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
5		epel 5433	. . 3 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
6	4, 5	anbi12i 629	. 2 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
7		epel 5433	. 2 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
8	3, 6, 7	3imtr3g 298	1 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111   class class class wbr 5030   E cep 5429   Or wor 5437   We wwe 5477

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-eprel 5430  df-po 5438  df-so 5439  df-we 5480

This theorem is referenced by:  wefrc  5513  ordelord  6181