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Theorem exmidfodomrlemeldju 7306
Description: Lemma for exmidfodomr 7311. A variant of djur 7170. (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 3192 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 ∈ 1o)
3 el1o 6522 . . . . . . . . 9 (𝑥 ∈ 1o𝑥 = ∅)
42, 3sylib 122 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 = ∅)
54fveq2d 5579 . . . . . . 7 ((𝜑𝑥𝐴) → (inl‘𝑥) = (inl‘∅))
65eqeq2d 2216 . . . . . 6 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) ↔ 𝐵 = (inl‘∅)))
76biimpd 144 . . . . 5 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
87rexlimdva 2622 . . . 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 5579 . . . . . . 7 ((𝜑𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅))
1413eqeq2d 2216 . . . . . 6 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) ↔ 𝐵 = (inr‘∅)))
1514biimpd 144 . . . . 5 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1615rexlimdva 2622 . . . 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 7170 . . 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 1372  wcel 2175  wrex 2484  wss 3165  c0 3459  cfv 5270  1oc1o 6494  cdju 7138  inlcinl 7146  inrcinr 7147
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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1st 6225  df-2nd 6226  df-1o 6501  df-dju 7139  df-inl 7148  df-inr 7149
This theorem is referenced by:  exmidfodomrlemr  7309
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