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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 1816 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
4 | dfcleq 2728 | . 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 35877 | . 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 1535 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 △ csymdif 4258 class class class wbr 5148 E cep 5588 ran crn 5690 ⊗ ctxp 35812 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-symdif 4259 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-1st 8013 df-2nd 8014 df-txp 35836 |
This theorem is referenced by: brbigcup 35880 brsingle 35899 brimage 35908 brcart 35914 brapply 35920 brcup 35921 brcap 35922 |
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