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Mirrors > Home > MPE Home > Th. List > Mathboxes > brtxpsd2 | Structured version Visualization version GIF version |
Description: Another common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 21-Apr-2014.) |
Ref | Expression |
---|---|
brtxpsd2.1 | ⊢ 𝐴 ∈ V |
brtxpsd2.2 | ⊢ 𝐵 ∈ V |
brtxpsd2.3 | ⊢ 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V))) |
brtxpsd2.4 | ⊢ 𝐴𝐶𝐵 |
Ref | Expression |
---|---|
brtxpsd2 | ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brtxpsd2.4 | . . 3 ⊢ 𝐴𝐶𝐵 | |
2 | brtxpsd2.3 | . . . . 5 ⊢ 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V))) | |
3 | 2 | breqi 5153 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵) |
4 | brdif 5200 | . . . 4 ⊢ (𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵 ↔ (𝐴𝐶𝐵 ∧ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵)) | |
5 | 3, 4 | bitri 275 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐵 ∧ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵)) |
6 | 1, 5 | mpbiran 709 | . 2 ⊢ (𝐴𝑅𝐵 ↔ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵) |
7 | brtxpsd2.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | brtxpsd2.2 | . . 3 ⊢ 𝐵 ∈ V | |
9 | 7, 8 | brtxpsd 35875 | . 2 ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
10 | 6, 9 | bitri 275 | 1 ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∀wal 1534 = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∖ cdif 3959 △ csymdif 4257 class class class wbr 5147 E cep 5587 ran crn 5689 ⊗ ctxp 35811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-symdif 4258 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-eprel 5588 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-fo 6568 df-fv 6570 df-1st 8012 df-2nd 8013 df-txp 35835 |
This theorem is referenced by: brtxpsd3 35877 |
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