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Theorem abnex 4375
 Description: Sufficient condition for a class abstraction to be a proper class. Lemma for snnex 4376 and pwnex 4377. See the comment of abnexg 4374. (Contributed by BJ, 2-May-2021.)
Assertion
Ref Expression
abnex (∀𝑥(𝐹𝑉𝑥𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)
Distinct variable groups:   𝑥,𝑦   𝑦,𝐹
Allowed substitution hints:   𝐹(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem abnex
StepHypRef Expression
1 vprc 4067 . 2 ¬ V ∈ V
2 alral 2481 . . 3 (∀𝑥(𝐹𝑉𝑥𝐹) → ∀𝑥 ∈ V (𝐹𝑉𝑥𝐹))
3 rexv 2707 . . . . . . 7 (∃𝑥 ∈ V 𝑦 = 𝐹 ↔ ∃𝑥 𝑦 = 𝐹)
43bicomi 131 . . . . . 6 (∃𝑥 𝑦 = 𝐹 ↔ ∃𝑥 ∈ V 𝑦 = 𝐹)
54abbii 2256 . . . . 5 {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹}
65eleq1i 2206 . . . 4 ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
76biimpi 119 . . 3 ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V)
8 abnexg 4374 . . 3 (∀𝑥 ∈ V (𝐹𝑉𝑥𝐹) → ({𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝐹} ∈ V → V ∈ V))
92, 7, 8syl2im 38 . 2 (∀𝑥(𝐹𝑉𝑥𝐹) → ({𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V → V ∈ V))
101, 9mtoi 654 1 (∀𝑥(𝐹𝑉𝑥𝐹) → ¬ {𝑦 ∣ ∃𝑥 𝑦 = 𝐹} ∈ V)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∀wral 2417  ∃wrex 2418  Vcvv 2689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-un 4362 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3081  df-ss 3088  df-sn 3537  df-uni 3744  df-iun 3822 This theorem is referenced by:  pwnex  4377
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