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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 4642 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 9044 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | endom 9000 | . . . . . . . 8 ⊢ ({𝐴} ≈ 1o → {𝐴} ≼ 1o) | |
4 | 1sdom2 9265 | . . . . . . . 8 ⊢ 1o ≺ 2o | |
5 | domsdomtr 9137 | . . . . . . . . 9 ⊢ (({𝐴} ≼ 1o ∧ 1o ≺ 2o) → {𝐴} ≺ 2o) | |
6 | sdomdom 9001 | . . . . . . . . 9 ⊢ ({𝐴} ≺ 2o → {𝐴} ≼ 2o) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (({𝐴} ≼ 1o ∧ 1o ≺ 2o) → {𝐴} ≼ 2o) |
8 | 3, 4, 7 | sylancl 585 | . . . . . . 7 ⊢ ({𝐴} ≈ 1o → {𝐴} ≼ 2o) |
9 | 2, 8 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≼ 2o) |
10 | 1, 9 | eqbrtrrid 5184 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴, 𝐴} ≼ 2o) |
11 | preq2 4739 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
12 | 11 | breq1d 5158 | . . . . 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 2736 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2o)) |
15 | 14 | adantrd 491 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o)) |
16 | pr2ne 10028 | . . . 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 9000 | . . 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 3022 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 {csn 4629 {cpr 4631 class class class wbr 5148 1oc1o 8480 2oc2o 8481 ≈ cen 8961 ≼ cdom 8962 ≺ csdm 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-1o 8487 df-2o 8488 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 |
This theorem is referenced by: (None) |
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