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

Theorem brtpos0 7447
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 7449. (Contributed by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
brtpos0 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))

Proof of Theorem brtpos0
StepHypRef Expression
1 brtpos2 7446 . 2 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴)))
2 ssun2 3853 . . . . 5 {∅} ⊆ (dom 𝐹 ∪ {∅})
3 0ex 4866 . . . . . 6 ∅ ∈ V
43snid 4284 . . . . 5 ∅ ∈ {∅}
52, 4sselii 3674 . . . 4 ∅ ∈ (dom 𝐹 ∪ {∅})
65biantrur 528 . . 3 ( {∅}𝐹𝐴 ↔ (∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴))
7 cnvsn0 5681 . . . . . 6 {∅} = ∅
87unieqi 4521 . . . . 5 {∅} =
9 uni0 4541 . . . . 5 ∅ = ∅
108, 9eqtri 2714 . . . 4 {∅} = ∅
1110breq1i 4735 . . 3 ( {∅}𝐹𝐴 ↔ ∅𝐹𝐴)
126, 11bitr3i 266 . 2 ((∅ ∈ (dom 𝐹 ∪ {∅}) ∧ {∅}𝐹𝐴) ↔ ∅𝐹𝐴)
131, 12syl6bb 276 1 (𝐴𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2071  cun 3646  c0 3991  {csn 4253   cuni 4512   class class class wbr 4728  ccnv 5185  dom cdm 5186  tpos ctpos 7439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1818  ax-5 1920  ax-6 1986  ax-7 2022  ax-8 2073  ax-9 2080  ax-10 2100  ax-11 2115  ax-12 2128  ax-13 2323  ax-ext 2672  ax-sep 4857  ax-nul 4865  ax-pow 4916  ax-pr 4979  ax-un 7034
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1567  df-ex 1786  df-nf 1791  df-sb 1979  df-eu 2543  df-mo 2544  df-clab 2679  df-cleq 2685  df-clel 2688  df-nfc 2823  df-ne 2865  df-ral 2987  df-rex 2988  df-rab 2991  df-v 3274  df-sbc 3510  df-dif 3651  df-un 3653  df-in 3655  df-ss 3662  df-nul 3992  df-if 4163  df-pw 4236  df-sn 4254  df-pr 4256  df-op 4260  df-uni 4513  df-br 4729  df-opab 4789  df-mpt 4806  df-id 5096  df-xp 5192  df-rel 5193  df-cnv 5194  df-co 5195  df-dm 5196  df-rn 5197  df-res 5198  df-ima 5199  df-iota 5932  df-fun 5971  df-fn 5972  df-fv 5977  df-tpos 7440
This theorem is referenced by:  reldmtpos  7448  brtpos  7449  tpostpos  7460
  Copyright terms: Public domain W3C validator