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Theorem exmidfodomrlemeldju 7307
Description: Lemma for exmidfodomr 7312. A variant of djur 7171. (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 3193 . . . . . . . . 9 ((𝜑𝑥𝐴) → 𝑥 ∈ 1o)
3 el1o 6523 . . . . . . . . 9 (𝑥 ∈ 1o𝑥 = ∅)
42, 3sylib 122 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝑥 = ∅)
54fveq2d 5580 . . . . . . 7 ((𝜑𝑥𝐴) → (inl‘𝑥) = (inl‘∅))
65eqeq2d 2217 . . . . . 6 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) ↔ 𝐵 = (inl‘∅)))
76biimpd 144 . . . . 5 ((𝜑𝑥𝐴) → (𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
87rexlimdva 2623 . . . 4 (𝜑 → (∃𝑥𝐴 𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
98imp 124 . . 3 ((𝜑 ∧ ∃𝑥𝐴 𝐵 = (inl‘𝑥)) → 𝐵 = (inl‘∅))
109orcd 735 . 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 5580 . . . . . . 7 ((𝜑𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅))
1413eqeq2d 2217 . . . . . 6 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) ↔ 𝐵 = (inr‘∅)))
1514biimpd 144 . . . . 5 ((𝜑𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1615rexlimdva 2623 . . . 4 (𝜑 → (∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
1716imp 124 . . 3 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → 𝐵 = (inr‘∅))
1817olcd 736 . 2 ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
19 exmidfodomrlemeldju.el . . 3 (𝜑𝐵 ∈ (𝐴 ⊔ 1o))
20 djur 7171 . . 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 800 1 (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 710   = wceq 1373  wcel 2176  wrex 2485  wss 3166  c0 3460  cfv 5271  1oc1o 6495  cdju 7139  inlcinl 7147  inrcinr 7148
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-2nd 6227  df-1o 6502  df-dju 7140  df-inl 7149  df-inr 7150
This theorem is referenced by:  exmidfodomrlemr  7310
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