eldju2ndr - Intuitionistic Logic Explorer

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Theorem eldju2ndr 7038

Description: The second component of an element of a disjoint union is an element of the right class of the disjoint union if its first component is not the empty set. (Contributed by AV, 26-Jun-2022.)

**Assertion**
Ref	Expression
eldju2ndr	⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1^st ‘𝑋) ≠ ∅) → (2^nd ‘𝑋) ∈ 𝐵)

**Proof of Theorem eldju2ndr**
Step	Hyp	Ref	Expression
1		df-dju 7003	. . . . 5 ⊢ (𝐴 ⊔ 𝐵) = (({∅} × 𝐴) ∪ ({1_o} × 𝐵))
2	1	eleq2i 2233	. . . 4 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ 𝑋 ∈ (({∅} × 𝐴) ∪ ({1_o} × 𝐵)))
3		elun 3263	. . . 4 ⊢ (𝑋 ∈ (({∅} × 𝐴) ∪ ({1_o} × 𝐵)) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1_o} × 𝐵)))
4	2, 3	bitri 183	. . 3 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) ↔ (𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1_o} × 𝐵)))
5		elxp6 6137	. . . . 5 ⊢ (𝑋 ∈ ({∅} × 𝐴) ↔ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {∅} ∧ (2^nd ‘𝑋) ∈ 𝐴)))
6		elsni 3594	. . . . . . 7 ⊢ ((1^st ‘𝑋) ∈ {∅} → (1^st ‘𝑋) = ∅)
7		eqneqall 2346	. . . . . . 7 ⊢ ((1^st ‘𝑋) = ∅ → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐵))
8	6, 7	syl 14	. . . . . 6 ⊢ ((1^st ‘𝑋) ∈ {∅} → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐵))
9	8	ad2antrl 482	. . . . 5 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {∅} ∧ (2^nd ‘𝑋) ∈ 𝐴)) → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐵))
10	5, 9	sylbi 120	. . . 4 ⊢ (𝑋 ∈ ({∅} × 𝐴) → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐵))
11		elxp6 6137	. . . . 5 ⊢ (𝑋 ∈ ({1_o} × 𝐵) ↔ (𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {1_o} ∧ (2^nd ‘𝑋) ∈ 𝐵)))
12		simprr 522	. . . . . 6 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {1_o} ∧ (2^nd ‘𝑋) ∈ 𝐵)) → (2^nd ‘𝑋) ∈ 𝐵)
13	12	a1d 22	. . . . 5 ⊢ ((𝑋 = ⟨(1^st ‘𝑋), (2^nd ‘𝑋)⟩ ∧ ((1^st ‘𝑋) ∈ {1_o} ∧ (2^nd ‘𝑋) ∈ 𝐵)) → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐵))
14	11, 13	sylbi 120	. . . 4 ⊢ (𝑋 ∈ ({1_o} × 𝐵) → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐵))
15	10, 14	jaoi 706	. . 3 ⊢ ((𝑋 ∈ ({∅} × 𝐴) ∨ 𝑋 ∈ ({1_o} × 𝐵)) → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐵))
16	4, 15	sylbi 120	. 2 ⊢ (𝑋 ∈ (𝐴 ⊔ 𝐵) → ((1^st ‘𝑋) ≠ ∅ → (2^nd ‘𝑋) ∈ 𝐵))
17	16	imp 123	1 ⊢ ((𝑋 ∈ (𝐴 ⊔ 𝐵) ∧ (1^st ‘𝑋) ≠ ∅) → (2^nd ‘𝑋) ∈ 𝐵)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∨ wo 698 = wceq 1343 ∈ wcel 2136 ≠ wne 2336 ∪ cun 3114 ∅c0 3409 {csn 3576 ⟨cop 3579 × cxp 4602 ‘cfv 5188 1^st c1st 6106 2^nd c2nd 6107 1_oc1o 6377 ⊔ cdju 7002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-v 2728 df-sbc 2952 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-iota 5153 df-fun 5190 df-fv 5196 df-1st 6108 df-2nd 6109 df-dju 7003

This theorem is referenced by: updjudhf 7044

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