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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 326 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
3 | 2 | albii 1737 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
4 | dfcleq 2604 | . 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 31172 | . 2 ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
10 | 3, 4, 9 | 3bitr4ri 292 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝐵 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∀wal 1473 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∖ cdif 3537 △ csymdif 3805 class class class wbr 4583 E cep 4947 ran crn 5039 ⊗ ctxp 31106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 |
This theorem is referenced by: brbigcup 31175 brsingle 31194 brimage 31203 brcart 31209 brapply 31215 brcup 31216 brcap 31217 |
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