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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 9263 | . 2 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ {𝑥 ∣ 𝑥 ∈ 𝐴} ≼ 1o) | |
2 | abid2 2867 | . . 3 ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴 | |
3 | 2 | breq1i 5150 | . 2 ⊢ ({𝑥 ∣ 𝑥 ∈ 𝐴} ≼ 1o ↔ 𝐴 ≼ 1o) |
4 | 1, 3 | bitri 275 | 1 ⊢ (∃*𝑥 𝑥 ∈ 𝐴 ↔ 𝐴 ≼ 1o) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∈ wcel 2099 ∃*wmo 2528 {cab 2705 class class class wbr 5143 1oc1o 8474 ≼ cdom 8956 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-1o 8481 df-en 8959 df-dom 8960 df-sdom 8961 |
This theorem is referenced by: f1omoALT 47905 isthinc2 48019 indthincALT 48050 |
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