Theorem wessep 4320

Description: A subset of a set well-ordered by set membership is well-ordered by set membership. (Contributed by Jim Kingdon, 30-Sep-2021.)

Assertion

Ref	Expression
wessep	⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → E We 𝐵)

Proof of Theorem wessep

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 2993	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
2		ssel 2993	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴))
3		ssel 2993	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐴))
4	1, 2, 3	3anim123d 1250	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)))
5	4	adantl 271	. . . . 5 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)))
6	5	imdistani 433	. . . 4 ⊢ ((( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)))
7		wetrep 4115	. . . . . 6 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
8	7	adantlr 460	. . . . 5 ⊢ ((( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9		epel 4047	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		epel 4047	. . . . . 6 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
11	9, 10	anbi12i 447	. . . . 5 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
12		epel 4047	. . . . 5 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
13	8, 11, 12	3imtr4g 203	. . . 4 ⊢ ((( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
14	6, 13	syl 14	. . 3 ⊢ ((( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
15	14	ralrimivvva 2444	. 2 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
16		zfregfr 4316	. . 3 ⊢ E Fr 𝐵
17		df-wetr 4089	. . 3 ⊢ ( E We 𝐵 ↔ ( E Fr 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
18	16, 17	mpbiran 881	. 2 ⊢ ( E We 𝐵 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
19	15, 18	sylibr 132	1 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → E We 𝐵)

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919   ∈ wcel 1433  ∀wral 2348   ⊆ wss 2973   class class class wbr 3785   E cep 4042   Fr wfr 4083   We wwe 4085

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-eprel 4044  df-frfor 4086  df-frind 4087  df-wetr 4089

This theorem is referenced by: (None)