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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 6254 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))) | |
2 | ssun2 3301 | . . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅}) | |
3 | 0ex 4132 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | 3 | snid 3625 | . . . . 5 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3154 | . . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅}) |
6 | 5 | biantrur 303 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)) |
7 | cnvsn0 5099 | . . . . . 6 ⊢ ◡{∅} = ∅ | |
8 | 7 | unieqi 3821 | . . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅ |
9 | uni0 3838 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
10 | 8, 9 | eqtri 2198 | . . . 4 ⊢ ∪ ◡{∅} = ∅ |
11 | 10 | breq1i 4012 | . . 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 2148 ∪ cun 3129 ∅c0 3424 {csn 3594 ∪ cuni 3811 class class class wbr 4005 ◡ccnv 4627 dom cdm 4628 tpos ctpos 6247 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-fv 5226 df-tpos 6248 |
This theorem is referenced by: reldmtpos 6256 tpostpos 6267 |
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