Theorem exmidfodomrlemeldju 7307

Description: Lemma for exmidfodomr 7312. A variant of djur 7171. (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 3193	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 1o)
3		el1o 6523	. . . . . . . . 9 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
4	2, 3	sylib 122	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 = ∅)
5	4	fveq2d 5580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (inl‘𝑥) = (inl‘∅))
6	5	eqeq2d 2217	. . . . . 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 2623	. . . 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 735	. 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 5580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 1o) → (inr‘𝑥) = (inr‘∅))
14	13	eqeq2d 2217	. . . . . 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 2623	. . . 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 736	. 2 ⊢ ((𝜑 ∧ ∃𝑥 ∈ 1o 𝐵 = (inr‘𝑥)) → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))
19		exmidfodomrlemeldju.el	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ⊔ 1o))
20		djur 7171	. . 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 800	1 ⊢ (𝜑 → (𝐵 = (inl‘∅) ∨ 𝐵 = (inr‘∅)))

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 710   = wceq 1373   ∈ wcel 2176  ∃wrex 2485   ⊆ wss 3166  ∅c0 3460  ‘cfv 5271  1_oc1o 6495   ⊔ cdju 7139  inlcinl 7147  inrcinr 7148

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1st 6226  df-2nd 6227  df-1o 6502  df-dju 7140  df-inl 7149  df-inr 7150

This theorem is referenced by:  exmidfodomrlemr  7310