Theorem exmidfodomrlemreseldju 7201

Description: Lemma for exmidfodomrlemrALT 7204. A variant of eldju 7069. (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 3157	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
3		el1o 6440	. . . . . . . . . 10 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
4	2, 3	sylib 122	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
5	4	fveq2d 5521	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑥) = ((inl ↾ 𝐴)‘∅))
6	5	eqeq2d 2189	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐵 = ((inl ↾ 𝐴)‘∅)))
7		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	4, 7	eqeltrrd 2255	. . . . . . . 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 2594	. . . 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 733	. 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 5521	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → ((inr ↾ 1o)‘𝑥) = ((inr ↾ 1o)‘∅))
18	17	eqeq2d 2189	. . . . . 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 2594	. . . 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 734	. 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 7069	. . 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 798	1 ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∃wrex 2456   ⊆ wss 3131  ∅c0 3424   ↾ cres 4630  ‘cfv 5218  1_oc1o 6412   ⊔ cdju 7038  inlcinl 7046  inrcinr 7047

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-1o 6419  df-dju 7039  df-inl 7048  df-inr 7049

This theorem is referenced by:  exmidfodomrlemrALT  7204