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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 8170. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
brtpos0 | ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brtpos2 8167 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))) | |
2 | ssun2 4137 | . . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅}) | |
3 | 0ex 5268 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | 3 | snid 4626 | . . . . 5 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3945 | . . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅}) |
6 | 5 | biantrur 532 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)) |
7 | cnvsn0 6166 | . . . . . 6 ⊢ ◡{∅} = ∅ | |
8 | 7 | unieqi 4882 | . . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅ |
9 | uni0 4900 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
10 | 8, 9 | eqtri 2761 | . . . 4 ⊢ ∪ ◡{∅} = ∅ |
11 | 10 | breq1i 5116 | . . 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 205 ∧ wa 397 ∈ wcel 2107 ∪ cun 3912 ∅c0 4286 {csn 4590 ∪ cuni 4869 class class class wbr 5109 ◡ccnv 5636 dom cdm 5637 tpos ctpos 8160 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-fv 6508 df-tpos 8161 |
This theorem is referenced by: reldmtpos 8169 brtpos 8170 tpostpos 8181 |
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