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Theorem elsnxpOLD 5647
Description: Obsolete proof of elsnxp 5646 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 5101 . . . 4 (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
2 df-rex 2914 . . . . . . . 8 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3 an13 839 . . . . . . . . 9 ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ (𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
43exbii 1771 . . . . . . . 8 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
52, 4bitr4i 267 . . . . . . 7 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
6 velsn 4171 . . . . . . . . . 10 (𝑥 ∈ {𝑋} ↔ 𝑥 = 𝑋)
76anbi1i 730 . . . . . . . . 9 ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩))
8 simpr 477 . . . . . . . . . 10 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑥, 𝑦⟩)
9 opeq1 4377 . . . . . . . . . . 11 (𝑥 = 𝑋 → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
109adantr 481 . . . . . . . . . 10 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
118, 10eqtrd 2655 . . . . . . . . 9 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
127, 11sylbi 207 . . . . . . . 8 ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
1312reximi 3007 . . . . . . 7 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
145, 13sylbir 225 . . . . . 6 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1514eximi 1759 . . . . 5 (∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑥𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
16 19.9v 1893 . . . . 5 (∃𝑥𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩ ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1715, 16sylib 208 . . . 4 (∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
181, 17sylbi 207 . . 3 (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1918adantl 482 . 2 ((𝑋𝑉𝑍 ∈ ({𝑋} × 𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
20 nfv 1840 . . . 4 𝑦 𝑋𝑉
21 nfre1 3001 . . . 4 𝑦𝑦𝐴 𝑍 = ⟨𝑋, 𝑦
2220, 21nfan 1825 . . 3 𝑦(𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
23 simpr 477 . . . . 5 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
24 snidg 4184 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
2524adantr 481 . . . . . . 7 ((𝑋𝑉𝑦𝐴) → 𝑋 ∈ {𝑋})
26 simpr 477 . . . . . . 7 ((𝑋𝑉𝑦𝐴) → 𝑦𝐴)
27 opelxp 5116 . . . . . . . 8 (⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑦𝐴))
2827biimpri 218 . . . . . . 7 ((𝑋 ∈ {𝑋} ∧ 𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
2925, 26, 28syl2anc 692 . . . . . 6 ((𝑋𝑉𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
3029adantr 481 . . . . 5 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
3123, 30eqeltrd 2698 . . . 4 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
3231adantllr 754 . . 3 ((((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) ∧ 𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
33 simpr 477 . . 3 ((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
3422, 32, 33r19.29af 3071 . 2 ((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
3519, 34impbida 876 1 (𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2909  {csn 4155  cop 4161   × cxp 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-opab 4684  df-xp 5090
This theorem is referenced by: (None)
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