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Theorem bj-xnex 32698
 Description: Lemma for snnex 6917 and bj-pwnex 32699. (Contributed by BJ, 2-May-2021.)
Assertion
Ref Expression
bj-xnex (∀𝑦(𝐴𝑉𝑦𝐴) → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐴
Allowed substitution hints:   𝐴(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem bj-xnex
StepHypRef Expression
1 nfa1 2025 . . . . 5 𝑦𝑦(𝐴𝑉𝑦𝐴)
2 nfe1 2024 . . . . . . 7 𝑦𝑦 𝑥 = 𝐴
32nfab 2765 . . . . . 6 𝑦{𝑥 ∣ ∃𝑦 𝑥 = 𝐴}
43nfuni 4408 . . . . 5 𝑦 {𝑥 ∣ ∃𝑦 𝑥 = 𝐴}
5 nfcv 2761 . . . . 5 𝑦V
6 vex 3189 . . . . . . 7 𝑦 ∈ V
762a1i 12 . . . . . 6 (∀𝑦(𝐴𝑉𝑦𝐴) → (𝑦 {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} → 𝑦 ∈ V))
8 nfcv 2761 . . . . . . . . . . 11 𝑥𝐴
9 nfv 1840 . . . . . . . . . . 11 𝑥 𝑦𝐴
10 eleq2 2687 . . . . . . . . . . . . 13 (𝑥 = 𝐴 → (𝑦𝑥𝑦𝐴))
1110biimprd 238 . . . . . . . . . . . 12 (𝑥 = 𝐴 → (𝑦𝐴𝑦𝑥))
12 19.8a 2049 . . . . . . . . . . . 12 (𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴)
1311, 12jctird 566 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑦𝐴 → (𝑦𝑥 ∧ ∃𝑦 𝑥 = 𝐴)))
148, 9, 13spcimegf 3273 . . . . . . . . . 10 (𝐴𝑉 → (𝑦𝐴 → ∃𝑥(𝑦𝑥 ∧ ∃𝑦 𝑥 = 𝐴)))
1514imp 445 . . . . . . . . 9 ((𝐴𝑉𝑦𝐴) → ∃𝑥(𝑦𝑥 ∧ ∃𝑦 𝑥 = 𝐴))
1615sps 2053 . . . . . . . 8 (∀𝑦(𝐴𝑉𝑦𝐴) → ∃𝑥(𝑦𝑥 ∧ ∃𝑦 𝑥 = 𝐴))
17 eluniab 4413 . . . . . . . 8 (𝑦 {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ↔ ∃𝑥(𝑦𝑥 ∧ ∃𝑦 𝑥 = 𝐴))
1816, 17sylibr 224 . . . . . . 7 (∀𝑦(𝐴𝑉𝑦𝐴) → 𝑦 {𝑥 ∣ ∃𝑦 𝑥 = 𝐴})
1918a1d 25 . . . . . 6 (∀𝑦(𝐴𝑉𝑦𝐴) → (𝑦 ∈ V → 𝑦 {𝑥 ∣ ∃𝑦 𝑥 = 𝐴}))
207, 19impbid 202 . . . . 5 (∀𝑦(𝐴𝑉𝑦𝐴) → (𝑦 {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ↔ 𝑦 ∈ V))
211, 4, 5, 20eqrd 3602 . . . 4 (∀𝑦(𝐴𝑉𝑦𝐴) → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} = V)
22 vprc 4756 . . . . 5 ¬ V ∈ V
2322a1i 11 . . . 4 (∀𝑦(𝐴𝑉𝑦𝐴) → ¬ V ∈ V)
2421, 23eqneltrd 2717 . . 3 (∀𝑦(𝐴𝑉𝑦𝐴) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V)
25 uniexg 6908 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V)
2624, 25nsyl 135 . 2 (∀𝑦(𝐴𝑉𝑦𝐴) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V)
27 df-nel 2894 . 2 ({𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∈ V)
2826, 27sylibr 224 1 (∀𝑦(𝐴𝑉𝑦𝐴) → {𝑥 ∣ ∃𝑦 𝑥 = 𝐴} ∉ V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987  {cab 2607   ∉ wnel 2893  Vcvv 3186  ∪ cuni 4402 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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-un 6902 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nel 2894  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-uni 4403 This theorem is referenced by:  bj-pwnex  32699
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