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Theorem elsnxpOLD 5581
Description: Obsolete proof of elsnxp 5580 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 5045 . . . 4 (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
2 df-rex 2901 . . . . . . . 8 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
3 an13 835 . . . . . . . . 9 ((𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ (𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
43exbii 1763 . . . . . . . 8 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) ↔ ∃𝑦(𝑦𝐴 ∧ (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩)))
52, 4bitr4i 265 . . . . . . 7 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)))
6 velsn 4140 . . . . . . . . . 10 (𝑥 ∈ {𝑋} ↔ 𝑥 = 𝑋)
76anbi1i 726 . . . . . . . . 9 ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩))
8 simpr 475 . . . . . . . . . 10 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑥, 𝑦⟩)
9 opeq1 4334 . . . . . . . . . . 11 (𝑥 = 𝑋 → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
109adantr 479 . . . . . . . . . 10 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ = ⟨𝑋, 𝑦⟩)
118, 10eqtrd 2643 . . . . . . . . 9 ((𝑥 = 𝑋𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
127, 11sylbi 205 . . . . . . . 8 ((𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
1312reximi 2993 . . . . . . 7 (∃𝑦𝐴 (𝑥 ∈ {𝑋} ∧ 𝑍 = ⟨𝑥, 𝑦⟩) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
145, 13sylbir 223 . . . . . 6 (∃𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1514eximi 1751 . . . . 5 (∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑥𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
16 19.9v 1882 . . . . 5 (∃𝑥𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩ ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1715, 16sylib 206 . . . 4 (∃𝑥𝑦(𝑍 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ {𝑋} ∧ 𝑦𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
181, 17sylbi 205 . . 3 (𝑍 ∈ ({𝑋} × 𝐴) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
1918adantl 480 . 2 ((𝑋𝑉𝑍 ∈ ({𝑋} × 𝐴)) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
20 nfv 1829 . . . 4 𝑦 𝑋𝑉
21 nfre1 2987 . . . 4 𝑦𝑦𝐴 𝑍 = ⟨𝑋, 𝑦
2220, 21nfan 1815 . . 3 𝑦(𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
23 simpr 475 . . . . 5 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 = ⟨𝑋, 𝑦⟩)
24 snidg 4152 . . . . . . . 8 (𝑋𝑉𝑋 ∈ {𝑋})
2524adantr 479 . . . . . . 7 ((𝑋𝑉𝑦𝐴) → 𝑋 ∈ {𝑋})
26 simpr 475 . . . . . . 7 ((𝑋𝑉𝑦𝐴) → 𝑦𝐴)
27 opelxp 5060 . . . . . . . 8 (⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑦𝐴))
2827biimpri 216 . . . . . . 7 ((𝑋 ∈ {𝑋} ∧ 𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
2925, 26, 28syl2anc 690 . . . . . 6 ((𝑋𝑉𝑦𝐴) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
3029adantr 479 . . . . 5 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → ⟨𝑋, 𝑦⟩ ∈ ({𝑋} × 𝐴))
3123, 30eqeltrd 2687 . . . 4 (((𝑋𝑉𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
3231adantllr 750 . . 3 ((((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) ∧ 𝑦𝐴) ∧ 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
33 simpr 475 . . 3 ((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩)
3422, 32, 33r19.29af 3057 . 2 ((𝑋𝑉 ∧ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩) → 𝑍 ∈ ({𝑋} × 𝐴))
3519, 34impbida 872 1 (𝑋𝑉 → (𝑍 ∈ ({𝑋} × 𝐴) ↔ ∃𝑦𝐴 𝑍 = ⟨𝑋, 𝑦⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wrex 2896  {csn 4124  cop 4130   × cxp 5026
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-opab 4638  df-xp 5034
This theorem is referenced by: (None)
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