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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 4573 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 8568 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | endom 8530 | . . . . . . . 8 ⊢ ({𝐴} ≈ 1o → {𝐴} ≼ 1o) | |
4 | 1sdom2 8711 | . . . . . . . 8 ⊢ 1o ≺ 2o | |
5 | domsdomtr 8646 | . . . . . . . . 9 ⊢ (({𝐴} ≼ 1o ∧ 1o ≺ 2o) → {𝐴} ≺ 2o) | |
6 | sdomdom 8531 | . . . . . . . . 9 ⊢ ({𝐴} ≺ 2o → {𝐴} ≼ 2o) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (({𝐴} ≼ 1o ∧ 1o ≺ 2o) → {𝐴} ≼ 2o) |
8 | 3, 4, 7 | sylancl 588 | . . . . . . 7 ⊢ ({𝐴} ≈ 1o → {𝐴} ≼ 2o) |
9 | 2, 8 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≼ 2o) |
10 | 1, 9 | eqbrtrrid 5094 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴, 𝐴} ≼ 2o) |
11 | preq2 4663 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
12 | 11 | breq1d 5068 | . . . . 5 ⊢ (𝐵 = 𝐴 → ({𝐴, 𝐵} ≼ 2o ↔ {𝐴, 𝐴} ≼ 2o)) |
13 | 10, 12 | syl5ibr 248 | . . . 4 ⊢ (𝐵 = 𝐴 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2o)) |
14 | 13 | eqcoms 2829 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2o)) |
15 | 14 | adantrd 494 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o)) |
16 | pr2ne 9425 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | |
17 | 16 | biimprd 250 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2o)) |
18 | endom 8530 | . . 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 3100 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {csn 4560 {cpr 4562 class class class wbr 5058 1oc1o 8089 2oc2o 8090 ≈ cen 8500 ≼ cdom 8501 ≺ csdm 8502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-1o 8096 df-2o 8097 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 |
This theorem is referenced by: (None) |
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