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Mirrors > Home > MPE Home > Th. List > en3 | Structured version Visualization version GIF version |
Description: A set equinumerous to ordinal 3 is a triple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
en3 | ⊢ (𝐴 ≈ 3𝑜 → ∃𝑥∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2onn 7720 | . 2 ⊢ 2𝑜 ∈ ω | |
2 | df-3o 7562 | . 2 ⊢ 3𝑜 = suc 2𝑜 | |
3 | en2 8196 | . 2 ⊢ ((𝐴 ∖ {𝑥}) ≈ 2𝑜 → ∃𝑦∃𝑧(𝐴 ∖ {𝑥}) = {𝑦, 𝑧}) | |
4 | tpass 4287 | . . . 4 ⊢ {𝑥, 𝑦, 𝑧} = ({𝑥} ∪ {𝑦, 𝑧}) | |
5 | 4 | enp1ilem 8194 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((𝐴 ∖ {𝑥}) = {𝑦, 𝑧} → 𝐴 = {𝑥, 𝑦, 𝑧})) |
6 | 5 | 2eximdv 1848 | . 2 ⊢ (𝑥 ∈ 𝐴 → (∃𝑦∃𝑧(𝐴 ∖ {𝑥}) = {𝑦, 𝑧} → ∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧})) |
7 | 1, 2, 3, 6 | enp1i 8195 | 1 ⊢ (𝐴 ≈ 3𝑜 → ∃𝑥∃𝑦∃𝑧 𝐴 = {𝑥, 𝑦, 𝑧}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∖ cdif 3571 {csn 4177 {cpr 4179 {ctp 4181 class class class wbr 4653 2𝑜c2o 7554 3𝑜c3o 7555 ≈ cen 7952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-1o 7560 df-2o 7561 df-3o 7562 df-er 7742 df-en 7956 df-fin 7959 |
This theorem is referenced by: en4 8198 hash3tr 13272 |
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