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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 340 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ (𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
3 | 2 | albii 1820 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥 ∈ 𝑋) ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
4 | dfcleq 2817 | . 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 33358 | . 2 ⊢ (𝐴𝑅𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐵 ↔ 𝑥𝑆𝐴)) |
10 | 3, 4, 9 | 3bitr4ri 306 | 1 ⊢ (𝐴𝑅𝐵 ↔ 𝐵 = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 △ csymdif 4220 class class class wbr 5068 E cep 5466 ran crn 5558 ⊗ ctxp 33293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-symdif 4221 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-eprel 5467 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-1st 7691 df-2nd 7692 df-txp 33317 |
This theorem is referenced by: brbigcup 33361 brsingle 33380 brimage 33389 brcart 33395 brapply 33401 brcup 33402 brcap 33403 |
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