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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 4138 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 7907 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1𝑜) | |
3 | endom 7868 | . . . . . . . 8 ⊢ ({𝐴} ≈ 1𝑜 → {𝐴} ≼ 1𝑜) | |
4 | 1sdom2 8044 | . . . . . . . 8 ⊢ 1𝑜 ≺ 2𝑜 | |
5 | domsdomtr 7980 | . . . . . . . . 9 ⊢ (({𝐴} ≼ 1𝑜 ∧ 1𝑜 ≺ 2𝑜) → {𝐴} ≺ 2𝑜) | |
6 | sdomdom 7869 | . . . . . . . . 9 ⊢ ({𝐴} ≺ 2𝑜 → {𝐴} ≼ 2𝑜) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (({𝐴} ≼ 1𝑜 ∧ 1𝑜 ≺ 2𝑜) → {𝐴} ≼ 2𝑜) |
8 | 3, 4, 7 | sylancl 693 | . . . . . . 7 ⊢ ({𝐴} ≈ 1𝑜 → {𝐴} ≼ 2𝑜) |
9 | 2, 8 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≼ 2𝑜) |
10 | 1, 9 | syl5eqbrr 4619 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴, 𝐴} ≼ 2𝑜) |
11 | preq2 4213 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
12 | 11 | breq1d 4593 | . . . . 5 ⊢ (𝐵 = 𝐴 → ({𝐴, 𝐵} ≼ 2𝑜 ↔ {𝐴, 𝐴} ≼ 2𝑜)) |
13 | 10, 12 | syl5ibr 235 | . . . 4 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2𝑜)) |
14 | 13 | eqcoms 2618 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2𝑜)) |
15 | 14 | adantrd 483 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2𝑜)) |
16 | pr2ne 8711 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2𝑜 ↔ 𝐴 ≠ 𝐵)) | |
17 | 16 | biimprd 237 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2𝑜)) |
18 | endom 7868 | . . 3 ⊢ ({𝐴, 𝐵} ≈ 2𝑜 → {𝐴, 𝐵} ≼ 2𝑜) | |
19 | 17, 18 | syl6com 36 | . 2 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2𝑜)) |
20 | 15, 19 | pm2.61ine 2865 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 {csn 4125 {cpr 4127 class class class wbr 4583 1𝑜c1o 7440 2𝑜c2o 7441 ≈ cen 7838 ≼ cdom 7839 ≺ csdm 7840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-om 6958 df-1o 7447 df-2o 7448 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 |
This theorem is referenced by: (None) |
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