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Theorem exmidfodomrlemeldju 7075
 Description: Lemma for exmidfodomr 7080. A variant of djur 6964. (Contributed by Jim Kingdon, 2-Jul-2022.)
Hypotheses
Ref Expression
exmidfodomrlemeldju.a (𝜑𝐴 ⊆ 1o)
exmidfodomrlemeldju.el (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
Assertion
Ref Expression
exmidfodomrlemeldju (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))

Proof of Theorem exmidfodomrlemeldju
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 exmidfodomrlemeldju.a . . . . . . . . . 10 (𝜑𝐴 ⊆ 1o)
21sselda 3103 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 ∈ 1o)
3 el1o 6343 . . . . . . . . 9 (𝑥 ∈ 1o𝑥 = ∅)
42, 3sylib 121 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 = ∅)
54fveq2d 5434 . . . . . . 7 ((𝜑𝑥𝐴) → (inl‘𝑥) = (inl‘∅))
65eqeq2d 2152 . . . . . 6 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) ↔ 𝐵 = (inl‘∅)))
76biimpd 143 . . . . 5 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
87rexlimdva 2553 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
98imp 123 . . 3 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = (inl‘𝑥)) → 𝐵 = (inl‘∅))
109orcd 723 . 2 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = (inl‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
11 simpr 109 . . . . . . . . 9 ((𝜑𝑥 ∈ 1o) → 𝑥 ∈ 1o)
1211, 3sylib 121 . . . . . . . 8 ((𝜑𝑥 ∈ 1o) → 𝑥 = ∅)
1312fveq2d 5434 . . . . . . 7 ((𝜑𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅))
1413eqeq2d 2152 . . . . . 6 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) ↔ 𝐵 = (inr‘∅)))
1514biimpd 143 . . . . 5 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1615rexlimdva 2553 . . . 4 (𝜑 → (∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1716imp 123 . . 3 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → 𝐵 = (inr‘∅))
1817olcd 724 . 2 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
19 exmidfodomrlemeldju.el . . 3 (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
20 djur 6964 . . 3 (𝐵 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑥𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)))
2119, 20sylib 121 . 2 (𝜑 → (∃𝑥𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)))
2210, 18, 21mpjaodan 788 1 (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1332   ∈ wcel 1481  ∃wrex 2418   ⊆ wss 3077  ∅c0 3369  ‘cfv 5132  1oc1o 6315   ⊔ cdju 6932  inlcinl 6940  inrcinr 6941 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4224  df-iord 4297  df-on 4299  df-suc 4302  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-1st 6047  df-2nd 6048  df-1o 6322  df-dju 6933  df-inl 6942  df-inr 6943 This theorem is referenced by:  exmidfodomrlemr  7078
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