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	⊢ (𝜑 → 𝐴 ⊆ 1o)
exmidfodomrlemeldju.el	⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))

Assertion

Ref	Expression
exmidfodomrlemeldju	⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))

Proof of Theorem exmidfodomrlemeldju

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidfodomrlemeldju.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ⊆ 1o)
2	1	sselda 3170	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
3		el1o 6463	. . . . . . . . 9 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
4	2, 3	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
5	4	fveq2d 5538	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (inl‘𝑥) = (inl‘∅))
6	5	eqeq2d 2201	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = (inl‘𝑥) ↔ 𝐵 = (inl‘∅)))
7	6	biimpd 144	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
8	7	rexlimdva 2607	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) → 𝐵 = (inl‘∅)))
9	8	imp 124	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥)) → 𝐵 = (inl‘∅))
10	9	orcd 734	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
11		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → 𝑥 ∈ 1o)
12	11, 3	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → 𝑥 = ∅)
13	12	fveq2d 5538	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅))
14	13	eqeq2d 2201	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) ↔ 𝐵 = (inr‘∅)))
15	14	biimpd 144	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
16	15	rexlimdva 2607	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥) → 𝐵 = (inr‘∅)))
17	16	imp 124	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → 𝐵 = (inr‘∅))
18	17	olcd 735	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
19		exmidfodomrlemeldju.el	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))
20		djur 7099	. . 3 ⊢ (𝐵 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)))
21	19, 20	sylib 122	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = (inl‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)))
22	10, 18, 21	mpjaodan 799	1 ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2160  ∃wrex 2469   ⊆ wss 3144  ∅c0 3437  ‘cfv 5235  1_oc1o 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