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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 (𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 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.
StepHypRef Expression
1 elxp 5055 . . . 4 (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
2 df-rex 2902 . . . . . . . 8 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3 an13 836 . . . . . . . . 9 ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ (𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
43exbii 1764 . . . . . . . 8 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
52, 4bitr4i 266 . . . . . . 7 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
6 velsn 4141 . . . . . . . . . 10 (𝑥 ∈ {𝑋} ↔ 𝑥 = 𝑋)
76anbi1i 727 . . . . . . . . 9 ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩))
8 simpr 476 . . . . . . . . . 10 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑥, 𝑦⟩)
9 opeq1 4340 . . . . . . . . . . 11 (𝑥 = 𝑋 → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
109adantr 480 . . . . . . . . . 10 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
118, 10eqtrd 2644 . . . . . . . . 9 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
127, 11sylbi 206 . . . . . . . 8 ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
1312reximi 2994 . . . . . . 7 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
145, 13sylbir 224 . . . . . 6 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1514eximi 1752 . . . . 5 (∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑥𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
16 19.9v 1883 . . . . 5 (∃𝑥𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩ ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1715, 16sylib 207 . . . 4 (∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
181, 17sylbi 206 . . 3 (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1918adantl 481 . 2 ((𝑋𝑉𝑍 ∈ ({𝑋} × 𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
20 nfv 1830 . . . 4 𝑦 𝑋𝑉
21 nfre1 2988 . . . 4 𝑦𝑦𝐴 𝑍 = ⟨𝑋, 𝑦
2220, 21nfan 1816 . . 3 𝑦(𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
23 simpr 476 . . . . 5 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
24 snidg 4153 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
2524adantr 480 . . . . . . 7 ((𝑋𝑉𝑦𝐴) → 𝑋 ∈ {𝑋})
26 simpr 476 . . . . . . 7 ((𝑋𝑉𝑦𝐴) → 𝑦𝐴)
27 opelxp 5070 . . . . . . . 8 (⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑦𝐴))
2827biimpri 217 . . . . . . 7 ((𝑋 ∈ {𝑋} ∧ 𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
2925, 26, 28syl2anc 691 . . . . . 6 ((𝑋𝑉𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
3029adantr 480 . . . . 5 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
3123, 30eqeltrd 2688 . . . 4 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
3231adantllr 751 . . 3 ((((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) ∧ 𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
33 simpr 476 . . 3 ((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
3422, 32, 33r19.29af 3058 . 2 ((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
3519, 34impbida 873 1 (𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
 Colors of variables: wff setvar class 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)
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