Theorem exmidfodomrlemreseldju 7378

Description: Lemma for exmidfodomrlemrALT 7381. A variant of eldju 7235. (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 3224	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
3		el1o 6583	. . . . . . . . . 10 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
4	2, 3	sylib 122	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
5	4	fveq2d 5631	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((inl ↾ 𝐴)‘𝑥) = ((inl ↾ 𝐴)‘∅))
6	5	eqeq2d 2241	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵 = ((inl ↾ 𝐴)‘𝑥) ↔ 𝐵 = ((inl ↾ 𝐴)‘∅)))
7		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8	4, 7	eqeltrrd 2307	. . . . . . . 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 2648	. . . 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 738	. 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 5631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → ((inr ↾ 1o)‘𝑥) = ((inr ↾ 1o)‘∅))
18	17	eqeq2d 2241	. . . . . 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 2648	. . . 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 739	. 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 7235	. . 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 803	1 ⊢ (𝜑 → ((∅ ∈ 𝐴 ∧ 𝐵 = ((inl ↾ 𝐴)‘∅)) ∨ 𝐵 = ((inr ↾ 1o)‘∅)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∃wrex 2509   ⊆ wss 3197  ∅c0 3491   ↾ cres 4721  ‘cfv 5318  1_oc1o 6555   ⊔ cdju 7204  inlcinl 7212  inrcinr 7213

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1st 6286  df-2nd 6287  df-1o 6562  df-dju 7205  df-inl 7214  df-inr 7215

This theorem is referenced by:  exmidfodomrlemrALT  7381