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Theorem exmidfodomrlemeldju 7229
Description: Lemma for exmidfodomr 7234. A variant of djur 7099. (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 3170 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 ∈ 1o)
3 el1o 6463 . . . . . . . . 9 (𝑥 ∈ 1o𝑥 = ∅)
42, 3sylib 122 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 = ∅)
54fveq2d 5538 . . . . . . 7 ((𝜑𝑥𝐴) → (inl‘𝑥) = (inl‘∅))
65eqeq2d 2201 . . . . . 6 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) ↔ 𝐵 = (inl‘∅)))
76biimpd 144 . . . . 5 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
87rexlimdva 2607 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
98imp 124 . . 3 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = (inl‘𝑥)) → 𝐵 = (inl‘∅))
109orcd 734 . 2 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = (inl‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
11 simpr 110 . . . . . . . . 9 ((𝜑𝑥 ∈ 1o) → 𝑥 ∈ 1o)
1211, 3sylib 122 . . . . . . . 8 ((𝜑𝑥 ∈ 1o) → 𝑥 = ∅)
1312fveq2d 5538 . . . . . . 7 ((𝜑𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅))
1413eqeq2d 2201 . . . . . 6 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) ↔ 𝐵 = (inr‘∅)))
1514biimpd 144 . . . . 5 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1615rexlimdva 2607 . . . 4 (𝜑 → (∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1716imp 124 . . 3 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → 𝐵 = (inr‘∅))
1817olcd 735 . 2 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
19 exmidfodomrlemeldju.el . . 3 (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
20 djur 7099 . . 3 (𝐵 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑥𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)))
2119, 20sylib 122 . 2 (𝜑 → (∃𝑥𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)))
2210, 18, 21mpjaodan 799 1 (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1364  wcel 2160  wrex 2469  wss 3144  c0 3437  cfv 5235  1oc1o 6435  cdju 7067  inlcinl 7075  inrcinr 7076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6166  df-2nd 6167  df-1o 6442  df-dju 7068  df-inl 7077  df-inr 7078
This theorem is referenced by:  exmidfodomrlemr  7232
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