Theorem exmidfodomrlemreseldju 7267

Description: Lemma for exmidfodomrlemrALT 7270. A variant of eldju 7134. (Contributed by Jim Kingdon, 9-Jul-2022.)

Hypotheses

Ref	Expression
exmidfodomrlemreseldju.a	⊢ (𝜑 → 𝐴 ⊆ 1o)
exmidfodomrlemreseldju.el	⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))

Assertion

Ref	Expression
exmidfodomrlemreseldju	⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Proof of Theorem exmidfodomrlemreseldju

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		exmidfodomrlemreseldju.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ⊆ 1o)
2	1	sselda 3183	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
3		el1o 6495	. . . . . . . . . 10 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
4	2, 3	sylib 122	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
5	4	fveq2d 5562	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑥) = ((inl ↾ 𝐴)‘∅))
6	5	eqeq2d 2208	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐵 = ((inl ↾ 𝐴)‘∅)))
7		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	4, 7	eqeltrrd 2274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∅ ∈ 𝐴)
9	8	biantrurd 305	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ((inl ↾ 𝐴)‘∅) ↔ (∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅))))
10	6, 9	bitrd 188	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ (∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅))))
11	10	biimpd 144	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) → (∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅))))
12	11	rexlimdva 2614	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) → (∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅))))
13	12	imp 124	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥)) → (∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)))
14	13	orcd 734	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥)) → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
15		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → 𝑥 ∈ 1o)
16	15, 3	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → 𝑥 = ∅)
17	16	fveq2d 5562	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → ((inr ↾ 1o)‘𝑥) = ((inr ↾ 1o)‘∅))
18	17	eqeq2d 2208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (𝐵 = ((inr ↾ 1o)‘𝑥) ↔ 𝐵 = ((inr ↾ 1o)‘∅)))
19	18	biimpd 144	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (𝐵 = ((inr ↾ 1o)‘𝑥) → 𝐵 = ((inr ↾ 1o)‘∅)))
20	19	rexlimdva 2614	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥) → 𝐵 = ((inr ↾ 1o)‘∅)))
21	20	imp 124	. . 3 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)) → 𝐵 = ((inr ↾ 1o)‘∅))
22	21	olcd 735	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)) → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))
23		exmidfodomrlemreseldju.el	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))
24		eldju 7134	. . 3 ⊢ (𝐵 ∈ (𝐴 ⊔ 1o) ↔ (∃𝑥 ∈ 𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)))
25	23, 24	sylib 122	. 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = ((inl ↾ 𝐴)‘𝑥) ∨ ∃𝑥 ∈ 1o 𝐵 = ((inr ↾ 1o)‘𝑥)))
26	14, 22, 25	mpjaodan 799	1 ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709   = wceq 1364   ∈ wcel 2167  ∃wrex 2476   ⊆ wss 3157  ∅c0 3450   ↾ cres 4665  ‘cfv 5258  1_oc1o 6467   ⊔ cdju 7103  inlcinl 7111  inrcinr 7112

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-1st 6198  df-2nd 6199  df-1o 6474  df-dju 7104  df-inl 7113  df-inr 7114

This theorem is referenced by:  exmidfodomrlemrALT  7270