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Theorem exmidfodomrlemeldju 7000
Description: Lemma for exmidfodomr 7005. A variant of djur 6904. (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 3061 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 ∈ 1o)
3 el1o 6286 . . . . . . . . 9 (𝑥 ∈ 1o𝑥 = ∅)
42, 3sylib 121 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 = ∅)
54fveq2d 5377 . . . . . . 7 ((𝜑𝑥𝐴) → (inl‘𝑥) = (inl‘∅))
65eqeq2d 2124 . . . . . 6 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) ↔ 𝐵 = (inl‘∅)))
76biimpd 143 . . . . 5 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
87rexlimdva 2521 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
98imp 123 . . 3 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = (inl‘𝑥)) → 𝐵 = (inl‘∅))
109orcd 705 . 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 5377 . . . . . . 7 ((𝜑𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅))
1413eqeq2d 2124 . . . . . 6 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) ↔ 𝐵 = (inr‘∅)))
1514biimpd 143 . . . . 5 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1615rexlimdva 2521 . . . 4 (𝜑 → (∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1716imp 123 . . 3 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → 𝐵 = (inr‘∅))
1817olcd 706 . 2 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
19 exmidfodomrlemeldju.el . . 3 (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
20 djur 6904 . . 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 770 1 (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wo 680   = wceq 1312  wcel 1461  wrex 2389  wss 3035  c0 3327  cfv 5079  1oc1o 6258  cdju 6872  inlcinl 6880  inrcinr 6881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-1st 5990  df-2nd 5991  df-1o 6265  df-dju 6873  df-inl 6882  df-inr 6883
This theorem is referenced by:  exmidfodomrlemr  7003
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