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Theorem brtpos0 8218
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 8220. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos0 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Proof of Theorem brtpos0
StepHypRef Expression
1 brtpos2 8217 . 2 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴)))
2 ssun2 4174 . . . . 5 {∅} ⊆ (dom 𝐹 ∪ {∅})
3 0ex 5308 . . . . . 6 ∅ ∈ V
43snid 4665 . . . . 5 ∅ ∈ {∅}
52, 4sselii 3980 . . . 4 ∅ ∈ (dom 𝐹 ∪ {∅})
65biantrur 532 . . 3 ( {∅}𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴))
7 cnvsn0 6210 . . . . . 6 {∅} = ∅
87unieqi 4922 . . . . 5 {∅} =
9 uni0 4940 . . . . 5 ∅ = ∅
108, 9eqtri 2761 . . . 4 {∅} = ∅
1110breq1i 5156 . . 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 205  wa 397  wcel 2107  cun 3947  c0 4323  {csn 4629   cuni 4909   class class class wbr 5149  ccnv 5676  dom cdm 5677  tpos ctpos 8210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-tpos 8211
This theorem is referenced by:  reldmtpos  8219  brtpos  8220  tpostpos  8231
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