Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
||
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 5045 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵) |
4 | brdif 5092 | . . . 4 ⊢ (𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵 ↔ (𝐴𝐶𝐵 ∧ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵)) | |
5 | 3, 4 | bitri 278 | . . 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 33882 | . 2 ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
10 | 6, 9 | bitri 278 | 1 ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 ∧ wa 399 ∀wal 1541 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∖ cdif 3850 △ csymdif 4142 class class class wbr 5039 E cep 5444 ran crn 5537 ⊗ ctxp 33818 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-symdif 4143 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-eprel 5445 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-1st 7739 df-2nd 7740 df-txp 33842 |
This theorem is referenced by: brtxpsd3 33884 |
Copyright terms: Public domain | W3C validator |