Theorem elsnxpOLD 5678

Description: Obsolete proof of elsnxp 5677 as of 14-Jul-2021. (Contributed by Thierry Arnoux, 10-Apr-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
elsnxpOLD	⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑉   𝑦,𝑋   𝑦,𝑍

Proof of Theorem elsnxpOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5131	. . . 4 ⊢ (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)))
2		df-rex 2918	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3		an13 840	. . . . . . . . 9 ⊢ ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
4	3	exbii 1774	. . . . . . . 8 ⊢ (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
5	2, 4	bitr4i 267	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)))
6		velsn 4193	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑋} ↔ 𝑥 = 𝑋)
7	6	anbi1i 731	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝑋 ∧ 𝑍 = ⟨𝑥, 𝑦⟩))
8		simpr 477	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑥, 𝑦⟩)
9		opeq1 4402	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
10	9	adantr 481	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
11	8, 10	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
12	7, 11	sylbi 207	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
13	12	reximi 3011	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
14	5, 13	sylbir 225	. . . . . 6 ⊢ (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
15	14	eximi 1762	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑥∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
16		19.9v 1896	. . . . 5 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩ ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
17	15, 16	sylib 208	. . . 4 ⊢ (∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
18	1, 17	sylbi 207	. . 3 ⊢ (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
19	18	adantl 482	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ ({𝑋} × 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
20		nfv 1843	. . . 4 ⊢ Ⅎ𝑦 𝑋 ∈ 𝑉
21		nfre1 3005	. . . 4 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩
22	20, 21	nfan 1828	. . 3 ⊢ Ⅎ𝑦(𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
23		simpr 477	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
24		snidg 4206	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
25	24	adantr 481	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → 𝑋 ∈ {𝑋})
26		simpr 477	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
27		opelxp 5146	. . . . . . . 8 ⊢ (⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴))
28	27	biimpri 218	. . . . . . 7 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
29	25, 26, 28	syl2anc 693	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
30	29	adantr 481	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
31	23, 30	eqeltrd 2701	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
32	31	adantllr 755	. . 3 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) ∧ 𝑦 ∈ 𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
33		simpr 477	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
34	22, 32, 33	r19.29af 3076	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
35	19, 34	impbida 877	1 ⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913  {csn 4177  ⟨cop 4183   × cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by: (None)