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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 4600 | . . . . . 6 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | ensn1g 8966 | . . . . . . 7 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
3 | endom 8922 | . . . . . . . 8 ⊢ ({𝐴} ≈ 1o → {𝐴} ≼ 1o) | |
4 | 1sdom2 9187 | . . . . . . . 8 ⊢ 1o ≺ 2o | |
5 | domsdomtr 9059 | . . . . . . . . 9 ⊢ (({𝐴} ≼ 1o ∧ 1o ≺ 2o) → {𝐴} ≺ 2o) | |
6 | sdomdom 8923 | . . . . . . . . 9 ⊢ ({𝐴} ≺ 2o → {𝐴} ≼ 2o) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (({𝐴} ≼ 1o ∧ 1o ≺ 2o) → {𝐴} ≼ 2o) |
8 | 3, 4, 7 | sylancl 587 | . . . . . . 7 ⊢ ({𝐴} ≈ 1o → {𝐴} ≼ 2o) |
9 | 2, 8 | syl 17 | . . . . . 6 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≼ 2o) |
10 | 1, 9 | eqbrtrrid 5142 | . . . . 5 ⊢ (𝐴 ∈ 𝐶 → {𝐴, 𝐴} ≼ 2o) |
11 | preq2 4696 | . . . . . 6 ⊢ (𝐵 = 𝐴 → {𝐴, 𝐵} = {𝐴, 𝐴}) | |
12 | 11 | breq1d 5116 | . . . . 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 2741 | . . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ 𝐶 → {𝐴, 𝐵} ≼ 2o)) |
15 | 14 | adantrd 493 | . 2 ⊢ (𝐴 = 𝐵 → ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴, 𝐵} ≼ 2o)) |
16 | pr2ne 9945 | . . . 4 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → ({𝐴, 𝐵} ≈ 2o ↔ 𝐴 ≠ 𝐵)) | |
17 | 16 | biimprd 248 | . . 3 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐴 ≠ 𝐵 → {𝐴, 𝐵} ≈ 2o)) |
18 | endom 8922 | . . 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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 {csn 4587 {cpr 4589 class class class wbr 5106 1oc1o 8406 2oc2o 8407 ≈ cen 8883 ≼ cdom 8884 ≺ csdm 8885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1o 8413 df-2o 8414 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 |
This theorem is referenced by: (None) |
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