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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 us to eliminate sethood hypotheses on 𝐴, 𝐵 in brtpos 7248. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
brtpos0 | ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brtpos2 7245 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴))) | |
2 | ssun2 3739 | . . . . 5 ⊢ {∅} ⊆ (◡dom 𝐹 ∪ {∅}) | |
3 | 0ex 4718 | . . . . . 6 ⊢ ∅ ∈ V | |
4 | 3 | snid 4155 | . . . . 5 ⊢ ∅ ∈ {∅} |
5 | 2, 4 | sselii 3565 | . . . 4 ⊢ ∅ ∈ (◡dom 𝐹 ∪ {∅}) |
6 | 5 | biantrur 526 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ (∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴)) |
7 | cnvsn0 5521 | . . . . . 6 ⊢ ◡{∅} = ∅ | |
8 | 7 | unieqi 4381 | . . . . 5 ⊢ ∪ ◡{∅} = ∪ ∅ |
9 | uni0 4401 | . . . . 5 ⊢ ∪ ∅ = ∅ | |
10 | 8, 9 | eqtri 2632 | . . . 4 ⊢ ∪ ◡{∅} = ∅ |
11 | 10 | breq1i 4590 | . . 3 ⊢ (∪ ◡{∅}𝐹𝐴 ↔ ∅𝐹𝐴) |
12 | 6, 11 | bitr3i 265 | . 2 ⊢ ((∅ ∈ (◡dom 𝐹 ∪ {∅}) ∧ ∪ ◡{∅}𝐹𝐴) ↔ ∅𝐹𝐴) |
13 | 1, 12 | syl6bb 275 | 1 ⊢ (𝐴 ∈ 𝑉 → (∅tpos 𝐹𝐴 ↔ ∅𝐹𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∈ wcel 1977 ∪ cun 3538 ∅c0 3874 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ◡ccnv 5037 dom cdm 5038 tpos ctpos 7238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-tpos 7239 |
This theorem is referenced by: reldmtpos 7247 brtpos 7248 tpostpos 7259 |
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