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Theorem wessep 4560
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 3141 . . . . . . 7 (𝐵𝐴 → (𝑥𝐵𝑥𝐴))
2 ssel 3141 . . . . . . 7 (𝐵𝐴 → (𝑦𝐵𝑦𝐴))
3 ssel 3141 . . . . . . 7 (𝐵𝐴 → (𝑧𝐵𝑧𝐴))
41, 2, 33anim123d 1314 . . . . . 6 (𝐵𝐴 → ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝐴𝑦𝐴𝑧𝐴)))
54adantl 275 . . . . 5 (( E We 𝐴𝐵𝐴) → ((𝑥𝐵𝑦𝐵𝑧𝐵) → (𝑥𝐴𝑦𝐴𝑧𝐴)))
65imdistani 443 . . . 4 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → (( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)))
7 wetrep 4343 . . . . . 6 (( E We 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
87adantlr 474 . . . . 5 ((( E We 𝐴𝐵𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
9 epel 4275 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
10 epel 4275 . . . . . 6 (𝑦 E 𝑧𝑦𝑧)
119, 10anbi12i 457 . . . . 5 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
12 epel 4275 . . . . 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 2553 . 2 (( E We 𝐴𝐵𝐴) → ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
16 zfregfr 4556 . . 3 E Fr 𝐵
17 df-wetr 4317 . . 3 ( E We 𝐵 ↔ ( E Fr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
1816, 17mpbiran 935 . 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 973  wcel 2141  wral 2448  wss 3121   class class class wbr 3987   E cep 4270   Fr wfr 4311   We wwe 4313
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-setind 4519
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-eprel 4272  df-frfor 4314  df-frind 4315  df-wetr 4317
This theorem is referenced by: (None)
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