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Theorem dfac5lem2 9527
Description: Lemma for dfac5 9531. (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 4824 . . 3 𝐴 = {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))}
32eleq2i 2903 . 2 (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ ⟨𝑤, 𝑔⟩ ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))})
4 eluniab 4826 . . 3 (⟨𝑤, 𝑔⟩ ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ↔ ∃𝑢(⟨𝑤, 𝑔⟩ ∈ 𝑢 ∧ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))))
5 r19.42v 3335 . . . . 5 (∃𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡)))
6 anass 472 . . . . 5 (((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡)) ↔ (⟨𝑤, 𝑔⟩ ∈ 𝑢 ∧ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))))
75, 6bitr2i 279 . . . 4 ((⟨𝑤, 𝑔⟩ ∈ 𝑢 ∧ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))) ↔ ∃𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)))
87exbii 1849 . . 3 (∃𝑢(⟨𝑤, 𝑔⟩ ∈ 𝑢 ∧ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))) ↔ ∃𝑢𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)))
9 rexcom4 3237 . . . 4 (∃𝑡𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ∃𝑢𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)))
10 df-rex 3132 . . . 4 (∃𝑡𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))))
119, 10bitr3i 280 . . 3 (∃𝑢𝑡 ((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))))
124, 8, 113bitri 300 . 2 (⟨𝑤, 𝑔⟩ ∈ {𝑢 ∣ (𝑢 ≠ ∅ ∧ ∃𝑡 𝑢 = ({𝑡} × 𝑡))} ↔ ∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))))
13 ancom 464 . . . . . . . . 9 (((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ (𝑢 = ({𝑡} × 𝑡) ∧ (⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅)))
14 ne0i 4273 . . . . . . . . . . 11 (⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅)
1514pm4.71i 563 . . . . . . . . . 10 (⟨𝑤, 𝑔⟩ ∈ 𝑢 ↔ (⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅))
1615anbi2i 625 . . . . . . . . 9 ((𝑢 = ({𝑡} × 𝑡) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑢) ↔ (𝑢 = ({𝑡} × 𝑡) ∧ (⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅)))
1713, 16bitr4i 281 . . . . . . . 8 (((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ (𝑢 = ({𝑡} × 𝑡) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑢))
1817exbii 1849 . . . . . . 7 (∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ∃𝑢(𝑢 = ({𝑡} × 𝑡) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑢))
19 snex 5305 . . . . . . . . 9 {𝑡} ∈ V
20 vex 3474 . . . . . . . . 9 𝑡 ∈ V
2119, 20xpex 7451 . . . . . . . 8 ({𝑡} × 𝑡) ∈ V
22 eleq2 2900 . . . . . . . 8 (𝑢 = ({𝑡} × 𝑡) → (⟨𝑤, 𝑔⟩ ∈ 𝑢 ↔ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡)))
2321, 22ceqsexv 3518 . . . . . . 7 (∃𝑢(𝑢 = ({𝑡} × 𝑡) ∧ ⟨𝑤, 𝑔⟩ ∈ 𝑢) ↔ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡))
2418, 23bitri 278 . . . . . 6 (∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡)) ↔ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡))
2524anbi2i 625 . . . . 5 ((𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))) ↔ (𝑡 ∧ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡)))
26 opelxp 5564 . . . . . . 7 (⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡) ↔ (𝑤 ∈ {𝑡} ∧ 𝑔𝑡))
27 velsn 4556 . . . . . . . . 9 (𝑤 ∈ {𝑡} ↔ 𝑤 = 𝑡)
28 equcom 2026 . . . . . . . . 9 (𝑤 = 𝑡𝑡 = 𝑤)
2927, 28bitri 278 . . . . . . . 8 (𝑤 ∈ {𝑡} ↔ 𝑡 = 𝑤)
3029anbi1i 626 . . . . . . 7 ((𝑤 ∈ {𝑡} ∧ 𝑔𝑡) ↔ (𝑡 = 𝑤𝑔𝑡))
3126, 30bitri 278 . . . . . 6 (⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡) ↔ (𝑡 = 𝑤𝑔𝑡))
3231anbi2i 625 . . . . 5 ((𝑡 ∧ ⟨𝑤, 𝑔⟩ ∈ ({𝑡} × 𝑡)) ↔ (𝑡 ∧ (𝑡 = 𝑤𝑔𝑡)))
33 an12 644 . . . . 5 ((𝑡 ∧ (𝑡 = 𝑤𝑔𝑡)) ↔ (𝑡 = 𝑤 ∧ (𝑡𝑔𝑡)))
3425, 32, 333bitri 300 . . . 4 ((𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))) ↔ (𝑡 = 𝑤 ∧ (𝑡𝑔𝑡)))
3534exbii 1849 . . 3 (∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))) ↔ ∃𝑡(𝑡 = 𝑤 ∧ (𝑡𝑔𝑡)))
36 vex 3474 . . . 4 𝑤 ∈ V
37 elequ1 2122 . . . . 5 (𝑡 = 𝑤 → (𝑡𝑤))
38 eleq2 2900 . . . . 5 (𝑡 = 𝑤 → (𝑔𝑡𝑔𝑤))
3937, 38anbi12d 633 . . . 4 (𝑡 = 𝑤 → ((𝑡𝑔𝑡) ↔ (𝑤𝑔𝑤)))
4036, 39ceqsexv 3518 . . 3 (∃𝑡(𝑡 = 𝑤 ∧ (𝑡𝑔𝑡)) ↔ (𝑤𝑔𝑤))
4135, 40bitri 278 . 2 (∃𝑡(𝑡 ∧ ∃𝑢((⟨𝑤, 𝑔⟩ ∈ 𝑢𝑢 ≠ ∅) ∧ 𝑢 = ({𝑡} × 𝑡))) ↔ (𝑤𝑔𝑤))
423, 12, 413bitri 300 1 (⟨𝑤, 𝑔⟩ ∈ 𝐴 ↔ (𝑤𝑔𝑤))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2115  {cab 2799  wne 3007  wrex 3127  c0 4266  {csn 4540  cop 4546   cuni 4811   × cxp 5526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-opab 5102  df-xp 5534  df-rel 5535
This theorem is referenced by:  dfac5lem5  9530
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