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

Theorem axunnd 9456
Description: A version of the Axiom of Union with no distinct variable conditions. (Contributed by NM, 2-Jan-2002.)
Assertion
Ref Expression
axunnd 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)

Proof of Theorem axunnd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 axunndlem1 9455 . . . 4 𝑤𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤)
2 nfnae 2351 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑦
3 nfnae 2351 . . . . . 6 𝑥 ¬ ∀𝑥 𝑥 = 𝑧
42, 3nfan 1868 . . . . 5 𝑥(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
5 nfnae 2351 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑦
6 nfnae 2351 . . . . . . 7 𝑦 ¬ ∀𝑥 𝑥 = 𝑧
75, 6nfan 1868 . . . . . 6 𝑦(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
8 nfv 1883 . . . . . . . 8 𝑤(¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧)
9 nfcvf 2817 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑦𝑥𝑦)
109adantr 480 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑦)
11 nfcvd 2794 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑤)
1210, 11nfeld 2802 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑦𝑤)
13 nfcvf 2817 . . . . . . . . . . 11 (¬ ∀𝑥 𝑥 = 𝑧𝑥𝑧)
1413adantl 481 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑥𝑧)
1511, 14nfeld 2802 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥 𝑤𝑧)
1612, 15nfand 1866 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(𝑦𝑤𝑤𝑧))
178, 16nfexd 2203 . . . . . . 7 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑤(𝑦𝑤𝑤𝑧))
1817, 12nfimd 1863 . . . . . 6 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤))
197, 18nfald 2201 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑥𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤))
20 nfcvd 2794 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑦𝑤)
21 nfcvf2 2818 . . . . . . . . . 10 (¬ ∀𝑥 𝑥 = 𝑦𝑦𝑥)
2221adantr 480 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → 𝑦𝑥)
2320, 22nfeqd 2801 . . . . . . . 8 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → Ⅎ𝑦 𝑤 = 𝑥)
247, 23nfan1 2106 . . . . . . 7 𝑦((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥)
25 elequ2 2044 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑦𝑤𝑦𝑥))
26 elequ1 2037 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝑤𝑧𝑥𝑧))
2725, 26anbi12d 747 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧)))
2827a1i 11 . . . . . . . . . 10 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → ((𝑦𝑤𝑤𝑧) ↔ (𝑦𝑥𝑥𝑧))))
294, 16, 28cbvexd 2314 . . . . . . . . 9 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
3029adantr 480 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∃𝑤(𝑦𝑤𝑤𝑧) ↔ ∃𝑥(𝑦𝑥𝑥𝑧)))
3125adantl 481 . . . . . . . 8 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (𝑦𝑤𝑦𝑥))
3230, 31imbi12d 333 . . . . . . 7 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → ((∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
3324, 32albid 2128 . . . . . 6 (((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) ∧ 𝑤 = 𝑥) → (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
3433ex 449 . . . . 5 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (𝑤 = 𝑥 → (∀𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))))
354, 19, 34cbvexd 2314 . . . 4 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → (∃𝑤𝑦(∃𝑤(𝑦𝑤𝑤𝑧) → 𝑦𝑤) ↔ ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
361, 35mpbii 223 . . 3 ((¬ ∀𝑥 𝑥 = 𝑦 ∧ ¬ ∀𝑥 𝑥 = 𝑧) → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
3736ex 449 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (¬ ∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)))
38 nfae 2349 . . . 4 𝑦𝑥 𝑥 = 𝑦
39 nfae 2349 . . . . . 6 𝑥𝑥 𝑥 = 𝑦
40 elirrv 8542 . . . . . . . . 9 ¬ 𝑦𝑦
41 elequ2 2044 . . . . . . . . 9 (𝑥 = 𝑦 → (𝑦𝑥𝑦𝑦))
4240, 41mtbiri 316 . . . . . . . 8 (𝑥 = 𝑦 → ¬ 𝑦𝑥)
4342intnanrd 983 . . . . . . 7 (𝑥 = 𝑦 → ¬ (𝑦𝑥𝑥𝑧))
4443sps 2093 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → ¬ (𝑦𝑥𝑥𝑧))
4539, 44nexd 2127 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
4645pm2.21d 118 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
4738, 46alrimi 2120 . . 3 (∀𝑥 𝑥 = 𝑦 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
48 19.8a 2090 . . 3 (∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥) → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
4947, 48syl 17 . 2 (∀𝑥 𝑥 = 𝑦 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
50 nfae 2349 . . . 4 𝑦𝑥 𝑥 = 𝑧
51 nfae 2349 . . . . . 6 𝑥𝑥 𝑥 = 𝑧
52 elirrv 8542 . . . . . . . . 9 ¬ 𝑧𝑧
53 elequ1 2037 . . . . . . . . 9 (𝑥 = 𝑧 → (𝑥𝑧𝑧𝑧))
5452, 53mtbiri 316 . . . . . . . 8 (𝑥 = 𝑧 → ¬ 𝑥𝑧)
5554intnand 982 . . . . . . 7 (𝑥 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
5655sps 2093 . . . . . 6 (∀𝑥 𝑥 = 𝑧 → ¬ (𝑦𝑥𝑥𝑧))
5751, 56nexd 2127 . . . . 5 (∀𝑥 𝑥 = 𝑧 → ¬ ∃𝑥(𝑦𝑥𝑥𝑧))
5857pm2.21d 118 . . . 4 (∀𝑥 𝑥 = 𝑧 → (∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
5950, 58alrimi 2120 . . 3 (∀𝑥 𝑥 = 𝑧 → ∀𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
6059, 48syl 17 . 2 (∀𝑥 𝑥 = 𝑧 → ∃𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥))
6137, 49, 60pm2.61ii 177 1 𝑥𝑦(∃𝑥(𝑦𝑥𝑥𝑧) → 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521  wex 1744  wnfc 2780
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-8 2032  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-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-eprel 5058  df-fr 5102
This theorem is referenced by:  zfcndun  9475  axunprim  31706
  Copyright terms: Public domain W3C validator