MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brtpos0 Structured version   Visualization version   GIF version

Theorem brtpos0 7597
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 us to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 7599. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos0 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Proof of Theorem brtpos0
StepHypRef Expression
1 brtpos2 7596 . 2 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴)))
2 ssun2 3975 . . . . 5 {∅} ⊆ (dom 𝐹 ∪ {∅})
3 0ex 4984 . . . . . 6 ∅ ∈ V
43snid 4400 . . . . 5 ∅ ∈ {∅}
52, 4sselii 3795 . . . 4 ∅ ∈ (dom 𝐹 ∪ {∅})
65biantrur 527 . . 3 ( {∅}𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴))
7 cnvsn0 5819 . . . . . 6 {∅} = ∅
87unieqi 4637 . . . . 5 {∅} =
9 uni0 4657 . . . . 5 ∅ = ∅
108, 9eqtri 2821 . . . 4 {∅} = ∅
1110breq1i 4850 . . 3 ( {∅}𝐹𝐴 ↔ ∅𝐹𝐴)
126, 11bitr3i 269 . 2 ((∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴) ↔ ∅𝐹𝐴)
131, 12syl6bb 279 1 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wcel 2157  cun 3767  c0 4115  {csn 4368   cuni 4628   class class class wbr 4843  ccnv 5311  dom cdm 5312  tpos ctpos 7589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109  df-tpos 7590
This theorem is referenced by:  reldmtpos  7598  brtpos  7599  tpostpos  7610
  Copyright terms: Public domain W3C validator