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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 5149 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵) |
| 4 | brdif 5196 | . . . 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 35895 | . 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 1538 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 △ csymdif 4252 class class class wbr 5143 E cep 5583 ran crn 5686 ⊗ ctxp 35831 |
| 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-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-symdif 4253 df-nul 4334 df-if 4526 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-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 |
| This theorem is referenced by: brtxpsd3 35897 |
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