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Theorem wessep 4462
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 3061 . . . . . . 7 (𝐵𝐴 → (𝑥𝐵𝑥𝐴))
2 ssel 3061 . . . . . . 7 (𝐵𝐴 → (𝑦𝐵𝑦𝐴))
3 ssel 3061 . . . . . . 7 (𝐵𝐴 → (𝑧𝐵𝑧𝐴))
41, 2, 33anim123d 1282 . . . . . 6 (𝐵𝐴 → ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝐴𝑦𝐴𝑧𝐴)))
54adantl 275 . . . . 5 (( E We 𝐴𝐵𝐴) → ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝐴𝑦𝐴𝑧𝐴)))
65imdistani 441 . . . 4 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)))
7 wetrep 4252 . . . . . 6 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
87adantlr 468 . . . . 5 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
9 epel 4184 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
10 epel 4184 . . . . . 6 (𝑦 E 𝑧𝑦𝑧)
119, 10anbi12i 455 . . . . 5 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
12 epel 4184 . . . . 5 (𝑥 E 𝑧𝑥𝑧)
138, 11, 123imtr4g 204 . . . 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 2492 . 2 (( E We 𝐴𝐵𝐴) → ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
16 zfregfr 4458 . . 3 E Fr 𝐵
17 df-wetr 4226 . . 3 ( E We 𝐵 ↔ ( E Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
1816, 17mpbiran 909 . 2 ( E We 𝐵 ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
1915, 18sylibr 133 1 (( E We 𝐴𝐵𝐴) → E We 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947  wcel 1465  wral 2393  wss 3041   class class class wbr 3899   E cep 4179   Fr wfr 4220   We wwe 4222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-eprel 4181  df-frfor 4223  df-frind 4224  df-wetr 4226
This theorem is referenced by: (None)
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