Theorem exmidfodomrlemreseldju 7516

Description: Lemma for exmidfodomrlemrALT 7519. A variant of eldju 7372. (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 3242	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
3		el1o 6683	. . . . . . . . . 10 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
4	2, 3	sylib 122	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
5	4	fveq2d 5679	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑥) = ((inl ↾ 𝐴)‘∅))
6	5	eqeq2d 2246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐵 = ((inl ↾ 𝐴)‘∅)))
7		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	4, 7	eqeltrrd 2312	. . . . . . . 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 2662	. . . 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 741	. 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 5679	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → ((inr ↾ 1o)‘𝑥) = ((inr ↾ 1o)‘∅))
18	17	eqeq2d 2246	. . . . . 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 2662	. . . 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 742	. 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 7372	. . 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 806	1 ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2205  ∃wrex 2523   ⊆ wss 3214  ∅c0 3512   ↾ cres 4756  ‘cfv 5357  1_oc1o 6653   ⊔ cdju 7341  inlcinl 7349  inrcinr 7350

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-1o 6660  df-dju 7342  df-inl 7351  df-inr 7352

This theorem is referenced by:  exmidfodomrlemrALT  7519