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Mirrors > Home > MPE Home > Th. List > prdom2 | Structured version Visualization version GIF version |
Description: An unordered pair has at most two elements. (Contributed by FL, 22-Feb-2011.) |
Ref | Expression |
---|---|
prdom2 | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 4381 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 8260 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1𝑜) | |
3 | endom 8222 | . . . . . . . 8 ⊢ ({𝐴} ≈ 1𝑜 → {𝐴} ≼ 1𝑜) | |
4 | 1sdom2 8401 | . . . . . . . 8 ⊢ 1𝑜 ≺ 2𝑜 | |
5 | domsdomtr 8337 | . . . . . . . . 9 ⊢ (({𝐴} ≼ 1𝑜 ∧ 1𝑜 ≺ 2𝑜) → {𝐴} ≺ 2𝑜) | |
6 | sdomdom 8223 | . . . . . . . . 9 ⊢ ({𝐴} ≺ 2𝑜 → {𝐴} ≼ 2𝑜) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (({𝐴} ≼ 1𝑜 ∧ 1𝑜 ≺ 2𝑜) → {𝐴} ≼ 2𝑜) |
8 | 3, 4, 7 | sylancl 581 | . . . . . . 7 ⊢ ({𝐴} ≈ 1𝑜 → {𝐴} ≼ 2𝑜) |
9 | 2, 8 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≼ 2𝑜) |
10 | 1, 9 | syl5eqbrr 4879 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴, 𝐴} ≼ 2𝑜) |
11 | preq2 4458 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
12 | 11 | breq1d 4853 | . . . . 5 ⊢ (𝐵 = 𝐴 → ({𝐴, 𝐵} ≼ 2𝑜 ↔ {𝐴, 𝐴} ≼ 2𝑜)) |
13 | 10, 12 | syl5ibr 238 | . . . 4 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2𝑜)) |
14 | 13 | eqcoms 2807 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2𝑜)) |
15 | 14 | adantrd 486 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2𝑜)) |
16 | pr2ne 9114 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2𝑜 ↔ 𝐴 ≠ 𝐵)) | |
17 | 16 | biimprd 240 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2𝑜)) |
18 | endom 8222 | . . 3 ⊢ ({𝐴, 𝐵} ≈ 2𝑜 → {𝐴, 𝐵} ≼ 2𝑜) | |
19 | 17, 18 | syl6com 37 | . 2 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2𝑜)) |
20 | 15, 19 | pm2.61ine 3054 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ≠ wne 2971 {csn 4368 {cpr 4370 class class class wbr 4843 1𝑜c1o 7792 2𝑜c2o 7793 ≈ cen 8192 ≼ cdom 8193 ≺ csdm 8194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-br 4844 df-opab 4906 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-om 7300 df-1o 7799 df-2o 7800 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 |
This theorem is referenced by: (None) |
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