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Theorem brtpos0 8257
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 8259. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos0 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Proof of Theorem brtpos0
StepHypRef Expression
1 brtpos2 8256 . 2 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴)))
2 ssun2 4189 . . . . 5 {∅} ⊆ (dom 𝐹 ∪ {∅})
3 0ex 5313 . . . . . 6 ∅ ∈ V
43snid 4667 . . . . 5 ∅ ∈ {∅}
52, 4sselii 3992 . . . 4 ∅ ∈ (dom 𝐹 ∪ {∅})
65biantrur 530 . . 3 ( {∅}𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴))
7 cnvsn0 6232 . . . . . 6 {∅} = ∅
87unieqi 4924 . . . . 5 {∅} =
9 uni0 4940 . . . . 5 ∅ = ∅
108, 9eqtri 2763 . . . 4 {∅} = ∅
1110breq1i 5155 . . 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 2106  cun 3961  c0 4339  {csn 4631   cuni 4912   class class class wbr 5148  ccnv 5688  dom cdm 5689  tpos ctpos 8249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-tpos 8250
This theorem is referenced by:  reldmtpos  8258  brtpos  8259  tpostpos  8270
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