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Theorem eldju2ndl 7365

Description: The second component of an element of a disjoint union is an element of the left class of the disjoint union if its first component is the empty set. (Contributed by AV, 26-Jun-2022.)

**Assertion**
Ref	Expression
eldju2ndl	⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1^st ‘𝑋) = ∅) → (2^nd ‘𝑋) ∈ 𝐴)

**Proof of Theorem eldju2ndl**
Step	Hyp	Ref	Expression
1		df-dju 7331	. . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1_o} × 𝐵))
2	1	eleq2i 2301	. . . 4 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1_o} × 𝐵)))
3		elun 3362	. . . 4 ⊢ (𝑋 ∈ (({∅} × 𝐴) ∪ ({1_o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1_o} × 𝐵)))
4	2, 3	bitri 184	. . 3 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1_o} × 𝐵)))
5		elxp6 6365	. . . . 5 ⊢ (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {∅} ∧ (2^nd ‘𝑋) ∈ 𝐴)))
6		simprr 533	. . . . . 6 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {∅} ∧ (2^nd ‘𝑋) ∈ 𝐴)) → (2^nd ‘𝑋) ∈ 𝐴)
7	6	a1d 22	. . . . 5 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {∅} ∧ (2^nd ‘𝑋) ∈ 𝐴)) → ((1^st ‘𝑋) = ∅ → (2^nd ‘𝑋) ∈ 𝐴))
8	5, 7	sylbi 121	. . . 4 ⊢ (𝑋 ∈ ({∅} × 𝐴) → ((1^st ‘𝑋) = ∅ → (2^nd ‘𝑋) ∈ 𝐴))
9		elxp6 6365	. . . . 5 ⊢ (𝑋 ∈ ({1_o} × 𝐵) ↔ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {1_o} ∧ (2^nd ‘𝑋) ∈ 𝐵)))
10		elsni 3709	. . . . . . 7 ⊢ ((1^st ‘𝑋) ∈ {1_o} → (1^st ‘𝑋) = 1_o)
11		1n0 6667	. . . . . . . 8 ⊢ 1_o ≠ ∅
12		neeq1 2427	. . . . . . . 8 ⊢ ((1^st ‘𝑋) = 1_o → ((1^st ‘𝑋) ≠ ∅ ↔ 1_o ≠ ∅))
13	11, 12	mpbiri 168	. . . . . . 7 ⊢ ((1^st ‘𝑋) = 1_o → (1^st ‘𝑋) ≠ ∅)
14		eqneqall 2424	. . . . . . . 8 ⊢ ((1^st ‘𝑋) = ∅ → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐴))
15	14	com12 30	. . . . . . 7 ⊢ ((1^st ‘𝑋) ≠ ∅ → ((1^st ‘𝑋) = ∅ → (2^nd ‘𝑋) ∈ 𝐴))
16	10, 13, 15	3syl 17	. . . . . 6 ⊢ ((1^st ‘𝑋) ∈ {1_o} → ((1^st ‘𝑋) = ∅ → (2^nd ‘𝑋) ∈ 𝐴))
17	16	ad2antrl 490	. . . . 5 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {1_o} ∧ (2^nd ‘𝑋) ∈ 𝐵)) → ((1^st ‘𝑋) = ∅ → (2^nd ‘𝑋) ∈ 𝐴))
18	9, 17	sylbi 121	. . . 4 ⊢ (𝑋 ∈ ({1_o} × 𝐵) → ((1^st ‘𝑋) = ∅ → (2^nd ‘𝑋) ∈ 𝐴))
19	8, 18	jaoi 724	. . 3 ⊢ ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1_o} × 𝐵)) → ((1^st ‘𝑋) = ∅ → (2^nd ‘𝑋) ∈ 𝐴))
20	4, 19	sylbi 121	. 2 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1^st ‘𝑋) = ∅ → (2^nd ‘𝑋) ∈ 𝐴))
21	20	imp 124	1 ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1^st ‘𝑋) = ∅) → (2^nd ‘𝑋) ∈ 𝐴)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∨ wo 716 = wceq 1398 ∈ wcel 2205 ≠ wne 2414 ∪ cun 3211 ∅c0 3510 {csn 3691 ⟨cop 3694 × cxp 4749 ‘cfv 5354 1^st c1st 6334 2^nd c2nd 6335 1_oc1o 6642 ⊔ cdju 7330

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 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556

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-ne 2415 df-ral 2527 df-rex 2528 df-v 2817 df-sbc 3045 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-br 4112 df-opab 4174 df-mpt 4175 df-id 4416 df-suc 4494 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-iota 5314 df-fun 5356 df-fv 5362 df-1st 6336 df-2nd 6337 df-1o 6649 df-dju 7331

This theorem is referenced by: updjudhf 7372 subctctexmid 16791

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