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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 4640 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 9015 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | endom 8971 | . . . . . . . 8 ⊢ ({𝐴} ≈ 1o → {𝐴} ≼ 1o) | |
4 | 1sdom2 9236 | . . . . . . . 8 ⊢ 1o ≺ 2o | |
5 | domsdomtr 9108 | . . . . . . . . 9 ⊢ (({𝐴} ≼ 1o ∧ 1o ≺ 2o) → {𝐴} ≺ 2o) | |
6 | sdomdom 8972 | . . . . . . . . 9 ⊢ ({𝐴} ≺ 2o → {𝐴} ≼ 2o) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (({𝐴} ≼ 1o ∧ 1o ≺ 2o) → {𝐴} ≼ 2o) |
8 | 3, 4, 7 | sylancl 586 | . . . . . . 7 ⊢ ({𝐴} ≈ 1o → {𝐴} ≼ 2o) |
9 | 2, 8 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≼ 2o) |
10 | 1, 9 | eqbrtrrid 5183 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴, 𝐴} ≼ 2o) |
11 | preq2 4737 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
12 | 11 | breq1d 5157 | . . . . 5 ⊢ (𝐵 = 𝐴 → ({𝐴, 𝐵} ≼ 2o ↔ {𝐴, 𝐴} ≼ 2o)) |
13 | 10, 12 | imbitrrid 245 | . . . 4 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2o)) |
14 | 13 | eqcoms 2740 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2o)) |
15 | 14 | adantrd 492 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o)) |
16 | pr2ne 9995 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | |
17 | 16 | biimprd 247 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2o)) |
18 | endom 8971 | . . 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 3025 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 {csn 4627 {cpr 4629 class class class wbr 5147 1oc1o 8455 2oc2o 8456 ≈ cen 8932 ≼ cdom 8933 ≺ csdm 8934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1o 8462 df-2o 8463 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 |
This theorem is referenced by: (None) |
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