MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac5lem2 Structured version   Visualization version   GIF version

Theorem dfac5lem2 9198
Description: Lemma for dfac5 9202. (Contributed by NM, 12-Apr-2004.)
Hypothesis
Ref Expression
dfac5lem.1 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
Assertion
Ref Expression
dfac5lem2 (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ (𝑤𝑔𝑤))
Distinct variable groups:   𝑤,𝑢,𝑡,,𝑔   𝑤,𝐴,𝑔
Allowed substitution hints:   𝐴(𝑢,𝑡,)

Proof of Theorem dfac5lem2
StepHypRef Expression
1 dfac5lem.1 . . . 4 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
21unieqi 4603 . . 3 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
32eleq2i 2836 . 2 (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ ⟨𝑤, 𝑔⟩ ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))})
4 eluniab 4605 . . 3 (⟨𝑤, 𝑔⟩ ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ↔ ∃𝑢(⟨𝑤, 𝑔⟩ ∈ 𝑢 ∧ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))))
5 r19.42v 3239 . . . . 5 (∃𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡)))
6 anass 460 . . . . 5 (((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡)) ↔ (⟨𝑤, 𝑔⟩ ∈ 𝑢 ∧ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))))
75, 6bitr2i 267 . . . 4 ((⟨𝑤, 𝑔⟩ ∈ 𝑢 ∧ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))) ↔ ∃𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)))
87exbii 1943 . . 3 (∃𝑢(⟨𝑤, 𝑔⟩ ∈ 𝑢 ∧ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))) ↔ ∃𝑢𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)))
9 rexcom4 3378 . . . 4 (∃𝑡𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ∃𝑢𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)))
10 df-rex 3061 . . . 4 (∃𝑡𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))))
119, 10bitr3i 268 . . 3 (∃𝑢𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))))
124, 8, 113bitri 288 . 2 (⟨𝑤, 𝑔⟩ ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ↔ ∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))))
13 ancom 452 . . . . . . . . 9 (((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ (𝑢 = ({𝑡} × 𝑡) ∧ (⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅)))
14 ne0i 4085 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅)
1514pm4.71i 555 . . . . . . . . . 10 (⟨𝑤, 𝑔⟩ ∈ 𝑢 ↔ (⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅))
1615anbi2i 616 . . . . . . . . 9 ((𝑢 = ({𝑡} × 𝑡) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑢) ↔ (𝑢 = ({𝑡} × 𝑡) ∧ (⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅)))
1713, 16bitr4i 269 . . . . . . . 8 (((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ (𝑢 = ({𝑡} × 𝑡) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑢))
1817exbii 1943 . . . . . . 7 (∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ∃𝑢(𝑢 = ({𝑡} × 𝑡) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑢))
19 snex 5064 . . . . . . . . 9 {𝑡} ∈ V
20 vex 3353 . . . . . . . . 9 𝑡 ∈ V
2119, 20xpex 7160 . . . . . . . 8 ({𝑡} × 𝑡) ∈ V
22 eleq2 2833 . . . . . . . 8 (𝑢 = ({𝑡} × 𝑡) → (⟨𝑤, 𝑔⟩ ∈ 𝑢 ↔ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡)))
2321, 22ceqsexv 3395 . . . . . . 7 (∃𝑢(𝑢 = ({𝑡} × 𝑡) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑢) ↔ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡))
2418, 23bitri 266 . . . . . 6 (∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡))
2524anbi2i 616 . . . . 5 ((𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))) ↔ (𝑡 ∧ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡)))
26 opelxp 5313 . . . . . . 7 (⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡) ↔ (𝑤 ∈ {𝑡} ∧ 𝑔𝑡))
27 velsn 4350 . . . . . . . . 9 (𝑤 ∈ {𝑡} ↔ 𝑤 = 𝑡)
28 equcom 2115 . . . . . . . . 9 (𝑤 = 𝑡𝑡 = 𝑤)
2927, 28bitri 266 . . . . . . . 8 (𝑤 ∈ {𝑡} ↔ 𝑡 = 𝑤)
3029anbi1i 617 . . . . . . 7 ((𝑤 ∈ {𝑡} ∧ 𝑔𝑡) ↔ (𝑡 = 𝑤𝑔𝑡))
3126, 30bitri 266 . . . . . 6 (⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡) ↔ (𝑡 = 𝑤𝑔𝑡))
3231anbi2i 616 . . . . 5 ((𝑡 ∧ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡)) ↔ (𝑡 ∧ (𝑡 = 𝑤𝑔𝑡)))
33 an12 635 . . . . 5 ((𝑡 ∧ (𝑡 = 𝑤𝑔𝑡)) ↔ (𝑡 = 𝑤 ∧ (𝑡𝑔𝑡)))
3425, 32, 333bitri 288 . . . 4 ((𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))) ↔ (𝑡 = 𝑤 ∧ (𝑡𝑔𝑡)))
3534exbii 1943 . . 3 (∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))) ↔ ∃𝑡(𝑡 = 𝑤 ∧ (𝑡𝑔𝑡)))
36 vex 3353 . . . 4 𝑤 ∈ V
37 elequ1 2162 . . . . 5 (𝑡 = 𝑤 → (𝑡𝑤))
38 eleq2 2833 . . . . 5 (𝑡 = 𝑤 → (𝑔𝑡𝑔𝑤))
3937, 38anbi12d 624 . . . 4 (𝑡 = 𝑤 → ((𝑡𝑔𝑡) ↔ (𝑤𝑔𝑤)))
4036, 39ceqsexv 3395 . . 3 (∃𝑡(𝑡 = 𝑤 ∧ (𝑡𝑔𝑡)) ↔ (𝑤𝑔𝑤))
4135, 40bitri 266 . 2 (∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))) ↔ (𝑤𝑔𝑤))
423, 12, 413bitri 288 1 (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ (𝑤𝑔𝑤))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wrex 3056  c0 4079  {csn 4334  cop 4340   cuni 4594   × cxp 5275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-opab 4872  df-xp 5283  df-rel 5284
This theorem is referenced by:  dfac5lem5  9201
  Copyright terms: Public domain W3C validator