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Mirrors > Home > MPE Home > Th. List > eliniseg2 | Structured version Visualization version GIF version |
Description: Eliminate the class existence constraint in eliniseg 5951. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 17-Nov-2015.) |
Ref | Expression |
---|---|
eliniseg2 | ⊢ (Rel 𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5960 | . . 3 ⊢ Rel ◡𝐴 | |
2 | elrelimasn 5946 | . . 3 ⊢ (Rel ◡𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶)) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐵◡𝐴𝐶) |
4 | relbrcnvg 5961 | . 2 ⊢ (Rel 𝐴 → (𝐵◡𝐴𝐶 ↔ 𝐶𝐴𝐵)) | |
5 | 3, 4 | syl5bb 284 | 1 ⊢ (Rel 𝐴 → (𝐶 ∈ (◡𝐴 “ {𝐵}) ↔ 𝐶𝐴𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∈ wcel 2105 {csn 4557 class class class wbr 5057 ◡ccnv 5547 “ cima 5551 Rel wrel 5553 |
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-pr 5320 |
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-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 |
This theorem is referenced by: isunit 19336 frege133d 39988 |
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