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Mirrors > Home > MPE Home > Th. List > modom2 | Structured version Visualization version GIF version |
Description: Two ways to express "at most one". (Contributed by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
modom2 | ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 ≼ 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | modom 8520 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ {𝑥 ∣ 𝑥 ∈ 𝐴} ≼ 1o) | |
2 | abid2 2911 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
3 | 2 | breq1i 4941 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ≼ 1o ↔ 𝐴 ≼ 1o) |
4 | 1, 3 | bitri 267 | 1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 ≼ 1o) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∈ wcel 2051 ∃*wmo 2549 {cab 2760 class class class wbr 4934 1oc1o 7904 ≼ cdom 8310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-tp 4449 df-op 4451 df-uni 4718 df-br 4935 df-opab 4997 df-tr 5036 df-id 5316 df-eprel 5321 df-po 5330 df-so 5331 df-fr 5370 df-we 5372 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-ord 6037 df-on 6038 df-lim 6039 df-suc 6040 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-om 7403 df-1o 7911 df-er 8095 df-en 8313 df-dom 8314 df-sdom 8315 |
This theorem is referenced by: (None) |
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