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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 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 3164 . . . . . . 7 (𝐵𝐴 → (𝑥𝐵𝑥𝐴))
2 ssel 3164 . . . . . . 7 (𝐵𝐴 → (𝑦𝐵𝑦𝐴))
3 ssel 3164 . . . . . . 7 (𝐵𝐴 → (𝑧𝐵𝑧𝐴))
41, 2, 33anim123d 1330 . . . . . 6 (𝐵𝐴 → ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝐴𝑦𝐴𝑧𝐴)))
54adantl 277 . . . . 5 (( E We 𝐴𝐵𝐴) → ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝐴𝑦𝐴𝑧𝐴)))
65imdistani 445 . . . 4 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)))
7 wetrep 4378 . . . . . 6 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
87adantlr 477 . . . . 5 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
9 epel 4310 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
10 epel 4310 . . . . . 6 (𝑦 E 𝑧𝑦𝑧)
119, 10anbi12i 460 . . . . 5 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
12 epel 4310 . . . . 5 (𝑥 E 𝑧𝑥𝑧)
138, 11, 123imtr4g 205 . . . 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 2573 . 2 (( E We 𝐴𝐵𝐴) → ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
16 zfregfr 4591 . . 3 E Fr 𝐵
17 df-wetr 4352 . . 3 ( E We 𝐵 ↔ ( E Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
1816, 17mpbiran 942 . 2 ( E We 𝐵 ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
1915, 18sylibr 134 1 (( E We 𝐴𝐵𝐴) → E We 𝐵)
Colors of variables: wff set class
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)
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