MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dtruALT2 Structured version   Visualization version   GIF version

Theorem dtruALT2 5335
Description: Alternate proof of dtru 5270 using ax-pr 5329 instead of ax-pow 5265. (Contributed by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dtruALT2 ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruALT2
StepHypRef Expression
1 0inp0 5258 . . . 4 (𝑦 = ∅ → ¬ 𝑦 = {∅})
2 snex 5331 . . . . 5 {∅} ∈ V
3 eqeq2 2833 . . . . . 6 (𝑥 = {∅} → (𝑦 = 𝑥𝑦 = {∅}))
43notbid 320 . . . . 5 (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅}))
52, 4spcev 3606 . . . 4 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥)
61, 5syl 17 . . 3 (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
7 0ex 5210 . . . 4 ∅ ∈ V
8 eqeq2 2833 . . . . 5 (𝑥 = ∅ → (𝑦 = 𝑥𝑦 = ∅))
98notbid 320 . . . 4 (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅))
107, 9spcev 3606 . . 3 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
116, 10pm2.61i 184 . 2 𝑥 ¬ 𝑦 = 𝑥
12 exnal 1823 . . 3 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥)
13 eqcom 2828 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
1413albii 1816 . . 3 (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
1512, 14xchbinx 336 . 2 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
1611, 15mpbi 232 1 ¬ ∀𝑥 𝑥 = 𝑦
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1531   = wceq 1533  wex 1776  c0 4290  {csn 4566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-v 3496  df-dif 3938  df-un 3940  df-nul 4291  df-sn 4567  df-pr 4569
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator