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Theorem brtpos0 8258
Description: The behavior of tpos when the left argument is the empty set (which is not an ordered pair but is the "default" value of an ordered pair when the arguments are proper classes). This allows to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 8260. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos0 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Proof of Theorem brtpos0
StepHypRef Expression
1 brtpos2 8257 . 2 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴)))
2 ssun2 4179 . . . . 5 {∅} ⊆ (dom 𝐹 ∪ {∅})
3 0ex 5307 . . . . . 6 ∅ ∈ V
43snid 4662 . . . . 5 ∅ ∈ {∅}
52, 4sselii 3980 . . . 4 ∅ ∈ (dom 𝐹 ∪ {∅})
65biantrur 530 . . 3 ( {∅}𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴))
7 cnvsn0 6230 . . . . . 6 {∅} = ∅
87unieqi 4919 . . . . 5 {∅} =
9 uni0 4935 . . . . 5 ∅ = ∅
108, 9eqtri 2765 . . . 4 {∅} = ∅
1110breq1i 5150 . . 3 ( {∅}𝐹𝐴 ↔ ∅𝐹𝐴)
126, 11bitr3i 277 . 2 ((∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴) ↔ ∅𝐹𝐴)
131, 12bitrdi 287 1 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2108  cun 3949  c0 4333  {csn 4626   cuni 4907   class class class wbr 5143  ccnv 5684  dom cdm 5685  tpos ctpos 8250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-tpos 8251
This theorem is referenced by:  reldmtpos  8259  brtpos  8260  tpostpos  8271  tposres0  48777
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