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Theorem int0OLD 4461
 Description: Obsolete proof of int0 4460 as of 26-Jul-2021. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
int0OLD ∅ = V

Proof of Theorem int0OLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 noel 3900 . . . . . 6 ¬ 𝑦 ∈ ∅
21pm2.21i 116 . . . . 5 (𝑦 ∈ ∅ → 𝑥𝑦)
32ax-gen 1719 . . . 4 𝑦(𝑦 ∈ ∅ → 𝑥𝑦)
4 equid 1936 . . . 4 𝑥 = 𝑥
53, 42th 254 . . 3 (∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦) ↔ 𝑥 = 𝑥)
65abbii 2736 . 2 {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)} = {𝑥𝑥 = 𝑥}
7 df-int 4446 . 2 ∅ = {𝑥 ∣ ∀𝑦(𝑦 ∈ ∅ → 𝑥𝑦)}
8 df-v 3191 . 2 V = {𝑥𝑥 = 𝑥}
96, 7, 83eqtr4i 2653 1 ∅ = V
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1478   = wceq 1480   ∈ wcel 1987  {cab 2607  Vcvv 3189  ∅c0 3896  ∩ cint 4445 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 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-v 3191  df-dif 3562  df-nul 3897  df-int 4446 This theorem is referenced by: (None)
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