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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 8260. (Contributed by Mario Carneiro, 10-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| brtpos0 | ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | brtpos2 8257 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))) | |
| 2 | ssun2 4179 | . . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅}) | |
| 3 | 0ex 5307 | . . . . . 6 ⊢ ∅ ∈ V | |
| 4 | 3 | snid 4662 | . . . . 5 ⊢ ∅ ∈ {∅} | 
| 5 | 2, 4 | sselii 3980 | . . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅}) | 
| 6 | 5 | biantrur 530 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)) | 
| 7 | cnvsn0 6230 | . . . . . 6 ⊢ ◡{∅} = ∅ | |
| 8 | 7 | unieqi 4919 | . . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅ | 
| 9 | uni0 4935 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
| 10 | 8, 9 | eqtri 2765 | . . . 4 ⊢ ∪ ◡{∅} = ∅ | 
| 11 | 10 | breq1i 5150 | . . 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 2108 ∪ cun 3949 ∅c0 4333 {csn 4626 ∪ cuni 4907 class class class wbr 5143 ◡ccnv 5684 dom cdm 5685 tpos ctpos 8250 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-tpos 8251 | 
| This theorem is referenced by: reldmtpos 8259 brtpos 8260 tpostpos 8271 tposres0 48777 | 
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