Theorem wessep 4723

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 3242	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑥 ∈ 𝐵 → 𝑥 ∈ 𝐴))
2		ssel 3242	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐴))
3		ssel 3242	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (𝑧 ∈ 𝐵 → 𝑧 ∈ 𝐴))
4	1, 2, 3	3anim123d 1360	. . . . . 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 449	. . . 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 4503	. . . . . 6 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
8	7	adantlr 481	. . . . 5 ⊢ ((( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9		epel 4435	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		epel 4435	. . . . . 6 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
11	9, 10	anbi12i 464	. . . . 5 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
12		epel 4435	. . . . 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 2633	. 2 ⊢ (( E We 𝐴 ∧ 𝐵 ⊆ 𝐴) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
16		zfregfr 4719	. . 3 ⊢ E Fr 𝐵
17		df-wetr 4477	. . 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 953	. 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 1009   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220   class class class wbr 4128   E cep 4430   Fr wfr 4471   We wwe 4473

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-opab 4191  df-eprel 4432  df-frfor 4474  df-frind 4475  df-wetr 4477

This theorem is referenced by: (None)