Theorem elsnxpOLD 5595

Description: Obsolete proof of elsnxp 5594 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 5055	. . . 4 ⊢ (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)))
2		df-rex 2902	. . . . . . . 8 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3		an13 836	. . . . . . . . 9 ⊢ ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
4	3	exbii 1764	. . . . . . . 8 ⊢ (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
5	2, 4	bitr4i 266	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)))
6		velsn 4141	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑋} ↔ 𝑥 = 𝑋)
7	6	anbi1i 727	. . . . . . . . 9 ⊢ ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝑋 ∧ 𝑍 = ⟨𝑥, 𝑦⟩))
8		simpr 476	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑥, 𝑦⟩)
9		opeq1 4340	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
10	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑋 ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
11	8, 10	eqtrd 2644	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
12	7, 11	sylbi 206	. . . . . . . 8 ⊢ ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
13	12	reximi 2994	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
14	5, 13	sylbir 224	. . . . . 6 ⊢ (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
15	14	eximi 1752	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑥∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
16		19.9v 1883	. . . . 5 ⊢ (∃𝑥∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩ ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
17	15, 16	sylib 207	. . . 4 ⊢ (∃𝑥∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
18	1, 17	sylbi 206	. . 3 ⊢ (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
19	18	adantl 481	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑍 ∈ ({𝑋} × 𝐴)) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
20		nfv 1830	. . . 4 ⊢ Ⅎ𝑦 𝑋 ∈ 𝑉
21		nfre1 2988	. . . 4 ⊢ Ⅎ𝑦∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩
22	20, 21	nfan 1816	. . 3 ⊢ Ⅎ𝑦(𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
23		simpr 476	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
24		snidg 4153	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
25	24	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → 𝑋 ∈ {𝑋})
26		simpr 476	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
27		opelxp 5070	. . . . . . . 8 ⊢ (⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴))
28	27	biimpri 217	. . . . . . 7 ⊢ ((𝑋 ∈ {𝑋} ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
29	25, 26, 28	syl2anc 691	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
30	29	adantr 480	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
31	23, 30	eqeltrd 2688	. . . 4 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
32	31	adantllr 751	. . 3 ⊢ ((((𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) ∧ 𝑦 ∈ 𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
33		simpr 476	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
34	22, 32, 33	r19.29af 3058	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
35	19, 34	impbida 873	1 ⊢ (𝑋 ∈ 𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑍 = ⟨𝑋, 𝑦⟩))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  {csn 4125  ⟨cop 4131   × cxp 5036

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044

This theorem is referenced by: (None)