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Mirrors > Home > MPE Home > Th. List > brtpos0 | Structured version Visualization version 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). This allows to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 8259. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
brtpos0 | ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brtpos2 8256 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))) | |
2 | ssun2 4189 | . . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅}) | |
3 | 0ex 5313 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | 3 | snid 4667 | . . . . 5 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3992 | . . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅}) |
6 | 5 | biantrur 530 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)) |
7 | cnvsn0 6232 | . . . . . 6 ⊢ ◡{∅} = ∅ | |
8 | 7 | unieqi 4924 | . . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅ |
9 | uni0 4940 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
10 | 8, 9 | eqtri 2763 | . . . 4 ⊢ ∪ ◡{∅} = ∅ |
11 | 10 | breq1i 5155 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ ∅𝐹𝐴) |
12 | 6, 11 | bitr3i 277 | . 2 ⊢ ((∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴) ↔ ∅𝐹𝐴) |
13 | 1, 12 | bitrdi 287 | 1 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2106 ∪ cun 3961 ∅c0 4339 {csn 4631 ∪ cuni 4912 class class class wbr 5148 ◡ccnv 5688 dom cdm 5689 tpos ctpos 8249 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-tpos 8250 |
This theorem is referenced by: reldmtpos 8258 brtpos 8259 tpostpos 8270 |
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