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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 339 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
3 | 2 | albii 1811 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
4 | dfcleq 2812 | . 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 33253 | . 2 ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
10 | 3, 4, 9 | 3bitr4ri 305 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝐵 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∀wal 1526 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 △ csymdif 4215 class class class wbr 5057 E cep 5457 ran crn 5549 ⊗ ctxp 33188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-symdif 4216 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-eprel 5458 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7678 df-2nd 7679 df-txp 33212 |
This theorem is referenced by: brbigcup 33256 brsingle 33275 brimage 33284 brcart 33290 brapply 33296 brcup 33297 brcap 33298 |
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