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Theorem phi011lem1 4598
 Description: Lemma for phi011 4599. (Contributed by SF, 3-Feb-2015.)
Assertion
Ref Expression
phi011lem1 (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → Phi A Phi B)

Proof of Theorem phi011lem1
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 ssun1 3426 . . . . 5 Phi A ( Phi A ∪ {0c})
21sseli 3269 . . . 4 (z Phi Az ( Phi A ∪ {0c}))
3 eleq2 2414 . . . 4 (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → (z ( Phi A ∪ {0c}) ↔ z ( Phi B ∪ {0c})))
42, 3syl5ib 210 . . 3 (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → (z Phi Az ( Phi B ∪ {0c})))
5 0cnelphi 4597 . . . . . 6 ¬ 0c Phi A
6 eleq1 2413 . . . . . 6 (z = 0c → (z Phi A ↔ 0c Phi A))
75, 6mtbiri 294 . . . . 5 (z = 0c → ¬ z Phi A)
87con2i 112 . . . 4 (z Phi A → ¬ z = 0c)
98a1i 10 . . 3 (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → (z Phi A → ¬ z = 0c))
10 elun 3220 . . . . . . 7 (z ( Phi B ∪ {0c}) ↔ (z Phi B z {0c}))
11 df-sn 3741 . . . . . . . . 9 {0c} = {z z = 0c}
1211abeq2i 2460 . . . . . . . 8 (z {0c} ↔ z = 0c)
1312orbi2i 505 . . . . . . 7 ((z Phi B z {0c}) ↔ (z Phi B z = 0c))
1410, 13bitri 240 . . . . . 6 (z ( Phi B ∪ {0c}) ↔ (z Phi B z = 0c))
1514biimpi 186 . . . . 5 (z ( Phi B ∪ {0c}) → (z Phi B z = 0c))
1615orcomd 377 . . . 4 (z ( Phi B ∪ {0c}) → (z = 0c z Phi B))
1716ord 366 . . 3 (z ( Phi B ∪ {0c}) → (¬ z = 0cz Phi B))
184, 9, 17ee22 1362 . 2 (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → (z Phi Az Phi B))
1918ssrdv 3278 1 (( Phi A ∪ {0c}) = ( Phi B ∪ {0c}) → Phi A Phi B)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   = wceq 1642   ∈ wcel 1710   ∪ cun 3207   ⊆ wss 3257  {csn 3737  0cc0c 4374   Phi cphi 4562 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-if 3663  df-sn 3741  df-int 3927  df-1c 4136  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565 This theorem is referenced by:  phi011  4599
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