![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > brtpos0 | GIF version |
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). (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
brtpos0 | ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brtpos2 6277 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))) | |
2 | ssun2 3314 | . . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅}) | |
3 | 0ex 4145 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | 3 | snid 3638 | . . . . 5 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3167 | . . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅}) |
6 | 5 | biantrur 303 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)) |
7 | cnvsn0 5115 | . . . . . 6 ⊢ ◡{∅} = ∅ | |
8 | 7 | unieqi 3834 | . . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅ |
9 | uni0 3851 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
10 | 8, 9 | eqtri 2210 | . . . 4 ⊢ ∪ ◡{∅} = ∅ |
11 | 10 | breq1i 4025 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ ∅𝐹𝐴) |
12 | 6, 11 | bitr3i 186 | . 2 ⊢ ((∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴) ↔ ∅𝐹𝐴) |
13 | 1, 12 | bitrdi 196 | 1 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2160 ∪ cun 3142 ∅c0 3437 {csn 3607 ∪ cuni 3824 class class class wbr 4018 ◡ccnv 4643 dom cdm 4644 tpos ctpos 6270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 df-tpos 6271 |
This theorem is referenced by: reldmtpos 6279 tpostpos 6290 |
Copyright terms: Public domain | W3C validator |