Theorem wessep 4595

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 3164	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
2		ssel 3164	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴))
3		ssel 3164	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐴))
4	1, 2, 3	3anim123d 1330	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)))
5	4	adantl 277	. . . . 5 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)))
6	5	imdistani 445	. . . 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 4378	. . . . . 6 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
8	7	adantlr 477	. . . . 5 ⊢ ((( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9		epel 4310	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		epel 4310	. . . . . 6 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
11	9, 10	anbi12i 460	. . . . 5 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
12		epel 4310	. . . . 5 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
13	8, 11, 12	3imtr4g 205	. . . 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 2573	. 2 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
16		zfregfr 4591	. . 3 ⊢ E Fr 𝐵
17		df-wetr 4352	. . 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 942	. 2 ⊢ ( E We 𝐵 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
19	15, 18	sylibr 134	1 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → E We 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   ∈ wcel 2160  ∀wral 2468   ⊆ wss 3144   class class class wbr 4018   E cep 4305   Fr wfr 4346   We wwe 4348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-eprel 4307  df-frfor 4349  df-frind 4350  df-wetr 4352

This theorem is referenced by: (None)