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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 5064 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵) |
4 | brdif 5111 | . . . 4 ⊢ (𝐴(𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V)))𝐵 ↔ (𝐴𝐶𝐵 ∧ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵)) | |
5 | 3, 4 | bitri 277 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝐶𝐵 ∧ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵)) |
6 | 1, 5 | mpbiran 707 | . 2 ⊢ (𝐴𝑅𝐵 ↔ ¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵) |
7 | brtxpsd2.1 | . . 3 ⊢ 𝐴 ∈ V | |
8 | brtxpsd2.2 | . . 3 ⊢ 𝐵 ∈ V | |
9 | 7, 8 | brtxpsd 33350 | . 2 ⊢ (¬ 𝐴ran ((V ⊗ E ) △ (𝑆 ⊗ V))𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
10 | 6, 9 | bitri 277 | 1 ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 ∧ wa 398 ∀wal 1531 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 △ csymdif 4217 class class class wbr 5058 E cep 5458 ran crn 5550 ⊗ ctxp 33286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-symdif 4218 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-eprel 5459 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-1st 7683 df-2nd 7684 df-txp 33310 |
This theorem is referenced by: brtxpsd3 33352 |
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