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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brtxpsd3 | Structured version Visualization version GIF version | ||
| Description: A third common abbreviation for quantifier-free definitions. (Contributed by Scott Fenton, 3-May-2014.) | 
| Ref | Expression | 
|---|---|
| brtxpsd2.1 | ⊢ 𝐴 ∈ V | 
| brtxpsd2.2 | ⊢ 𝐵 ∈ V | 
| brtxpsd2.3 | ⊢ 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V))) | 
| brtxpsd2.4 | ⊢ 𝐴𝐶𝐵 | 
| brtxpsd3.5 | ⊢ (𝑥 ∈ 𝑋 ↔ 𝑥𝑆𝐴) | 
| Ref | Expression | 
|---|---|
| brtxpsd3 | ⊢ (𝐴𝑅𝐵 ↔ 𝐵 = 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | brtxpsd3.5 | . . . 4 ⊢ (𝑥 ∈ 𝑋 ↔ 𝑥𝑆𝐴) | |
| 2 | 1 | bibi2i 337 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) | 
| 3 | 2 | albii 1819 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) | 
| 4 | dfcleq 2730 | . 2 ⊢ (𝐵 = 𝑋 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋)) | |
| 5 | brtxpsd2.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 6 | brtxpsd2.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 7 | brtxpsd2.3 | . . 3 ⊢ 𝑅 = (𝐶 ∖ ran ((V ⊗ E ) △ (𝑆 ⊗ V))) | |
| 8 | brtxpsd2.4 | . . 3 ⊢ 𝐴𝐶𝐵 | |
| 9 | 5, 6, 7, 8 | brtxpsd2 35896 | . 2 ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) | 
| 10 | 3, 4, 9 | 3bitr4ri 304 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝐵 = 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀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: brbigcup 35899 brsingle 35918 brimage 35927 brcart 35933 brapply 35939 brcup 35940 brcap 35941 | 
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