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

Theorem dtruALT 4929
 Description: Alternate proof of dtru 4887 which requires more axioms but is shorter and may be easier to understand. Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, theorem spcev 3331 requires that 𝑥 must not occur in the subexpression ¬ 𝑦 = {∅} in step 4 nor in the subexpression ¬ 𝑦 = ∅ in step 9. The proof verifier will require that 𝑥 and 𝑦 be in a distinct variable group to ensure this. You can check this by deleting the \$d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dtruALT ¬ ∀𝑥 𝑥 = 𝑦
Distinct variable group:   𝑥,𝑦

Proof of Theorem dtruALT
StepHypRef Expression
1 0inp0 4867 . . . 4 (𝑦 = ∅ → ¬ 𝑦 = {∅})
2 p0ex 4883 . . . . 5 {∅} ∈ V
3 eqeq2 2662 . . . . . 6 (𝑥 = {∅} → (𝑦 = 𝑥𝑦 = {∅}))
43notbid 307 . . . . 5 (𝑥 = {∅} → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = {∅}))
52, 4spcev 3331 . . . 4 𝑦 = {∅} → ∃𝑥 ¬ 𝑦 = 𝑥)
61, 5syl 17 . . 3 (𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
7 0ex 4823 . . . 4 ∅ ∈ V
8 eqeq2 2662 . . . . 5 (𝑥 = ∅ → (𝑦 = 𝑥𝑦 = ∅))
98notbid 307 . . . 4 (𝑥 = ∅ → (¬ 𝑦 = 𝑥 ↔ ¬ 𝑦 = ∅))
107, 9spcev 3331 . . 3 𝑦 = ∅ → ∃𝑥 ¬ 𝑦 = 𝑥)
116, 10pm2.61i 176 . 2 𝑥 ¬ 𝑦 = 𝑥
12 exnal 1794 . . 3 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑦 = 𝑥)
13 eqcom 2658 . . . 4 (𝑦 = 𝑥𝑥 = 𝑦)
1413albii 1787 . . 3 (∀𝑥 𝑦 = 𝑥 ↔ ∀𝑥 𝑥 = 𝑦)
1512, 14xchbinx 323 . 2 (∃𝑥 ¬ 𝑦 = 𝑥 ↔ ¬ ∀𝑥 𝑥 = 𝑦)
1611, 15mpbi 220 1 ¬ ∀𝑥 𝑥 = 𝑦
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∀wal 1521   = wceq 1523  ∃wex 1744  ∅c0 3948  {csn 4210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator