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Theorem exmidfodomrlemreseldju 7068
Description: Lemma for exmidfodomrlemrALT 7071. A variant of eldju 6956. (Contributed by Jim Kingdon, 9-Jul-2022.)
Hypotheses
Ref Expression
exmidfodomrlemreseldju.a (𝜑𝐴 ⊆ 1o)
exmidfodomrlemreseldju.el (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
Assertion
Ref Expression
exmidfodomrlemreseldju (𝜑 → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Proof of Theorem exmidfodomrlemreseldju
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 exmidfodomrlemreseldju.a . . . . . . . . . . 11 (𝜑𝐴 ⊆ 1o)
21sselda 3097 . . . . . . . . . 10 ((𝜑𝑥𝐴) → 𝑥 ∈ 1o)
3 el1o 6337 . . . . . . . . . 10 (𝑥 ∈ 1o𝑥 = ∅)
42, 3sylib 121 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 = ∅)
54fveq2d 5428 . . . . . . . 8 ((𝜑𝑥𝐴) → ((inl ↾ 𝐴)‘𝑥) = ((inl ↾ 𝐴)‘∅))
65eqeq2d 2151 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐵 = ((inl ↾ 𝐴)‘∅)))
7 simpr 109 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥𝐴)
84, 7eqeltrrd 2217 . . . . . . . 8 ((𝜑𝑥𝐴) → ∅ ∈ 𝐴)
98biantrurd 303 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐵 = ((inl ↾ 𝐴)‘∅) ↔ (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅))))
106, 9bitrd 187 . . . . . 6 ((𝜑𝑥𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅))))
1110biimpd 143 . . . . 5 ((𝜑𝑥𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) → (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅))))
1211rexlimdva 2549 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) → (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅))))
1312imp 123 . . 3 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥)) → (∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)))
1413orcd 722 . 2 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥)) → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
15 simpr 109 . . . . . . . . 9 ((𝜑𝑥 ∈ 1o) → 𝑥 ∈ 1o)
1615, 3sylib 121 . . . . . . . 8 ((𝜑𝑥 ∈ 1o) → 𝑥 = ∅)
1716fveq2d 5428 . . . . . . 7 ((𝜑𝑥 ∈ 1o) → ((inr ↾ 1o)‘𝑥) = ((inr ↾ 1o)‘∅))
1817eqeq2d 2151 . . . . . 6 ((𝜑𝑥 ∈ 1o) → (𝐵 = ((inr ↾ 1o)‘𝑥) ↔ 𝐵 = ((inr ↾ 1o)‘∅)))
1918biimpd 143 . . . . 5 ((𝜑𝑥 ∈ 1o) → (𝐵 = ((inr ↾ 1o)‘𝑥) → 𝐵 = ((inr ↾ 1o)‘∅)))
2019rexlimdva 2549 . . . 4 (𝜑 → (∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥) → 𝐵 = ((inr ↾ 1o)‘∅)))
2120imp 123 . . 3 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)) → 𝐵 = ((inr ↾ 1o)‘∅))
2221olcd 723 . 2 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)) → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
23 exmidfodomrlemreseldju.el . . 3 (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
24 eldju 6956 . . 3 (𝐵 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)))
2523, 24sylib 121 . 2 (𝜑 → (∃𝑥𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)))
2614, 22, 25mpjaodan 787 1 (𝜑 → ((∅ ∈ 𝐴𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 697   = wceq 1331  wcel 1480  wrex 2417  wss 3071  c0 3363  cres 4544  cfv 5126  1oc1o 6309  cdju 6925  inlcinl 6933  inrcinr 6934
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-id 4218  df-iord 4291  df-on 4293  df-suc 4296  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-1st 6041  df-2nd 6042  df-1o 6316  df-dju 6926  df-inl 6935  df-inr 6936
This theorem is referenced by:  exmidfodomrlemrALT  7071
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