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Theorem wessep 4356
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 𝐴𝐵𝐴) → E We 𝐵)

Proof of Theorem wessep
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3004 . . . . . . 7 (𝐵𝐴 → (𝑥𝐵𝑥𝐴))
2 ssel 3004 . . . . . . 7 (𝐵𝐴 → (𝑦𝐵𝑦𝐴))
3 ssel 3004 . . . . . . 7 (𝐵𝐴 → (𝑧𝐵𝑧𝐴))
41, 2, 33anim123d 1251 . . . . . 6 (𝐵𝐴 → ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝐴𝑦𝐴𝑧𝐴)))
54adantl 271 . . . . 5 (( E We 𝐴𝐵𝐴) → ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝐴𝑦𝐴𝑧𝐴)))
65imdistani 434 . . . 4 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)))
7 wetrep 4151 . . . . . 6 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
87adantlr 461 . . . . 5 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
9 epel 4083 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
10 epel 4083 . . . . . 6 (𝑦 E 𝑧𝑦𝑧)
119, 10anbi12i 448 . . . . 5 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
12 epel 4083 . . . . 5 (𝑥 E 𝑧𝑥𝑧)
138, 11, 123imtr4g 203 . . . 4 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
146, 13syl 14 . . 3 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
1514ralrimivvva 2450 . 2 (( E We 𝐴𝐵𝐴) → ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
16 zfregfr 4352 . . 3 E Fr 𝐵
17 df-wetr 4125 . . 3 ( E We 𝐵 ↔ ( E Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
1816, 17mpbiran 882 . 2 ( E We 𝐵 ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
1915, 18sylibr 132 1 (( E We 𝐴𝐵𝐴) → E We 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  w3a 920  wcel 1434  wral 2353  wss 2984   class class class wbr 3811   E cep 4078   Fr wfr 4119   We wwe 4121
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-setind 4316
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-eprel 4080  df-frfor 4122  df-frind 4123  df-wetr 4125
This theorem is referenced by: (None)
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