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Mirrors > Home > MPE Home > Th. List > hashdmpropge2 | Structured version Visualization version GIF version |
Description: The size of the domain of a class which contains two ordered pairs with different first components is greater than or equal to 2. (Contributed by AV, 12-Nov-2021.) |
Ref | Expression |
---|---|
hashdmpropge2.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
hashdmpropge2.b | ⊢ (𝜑 → 𝐵 ∈ 𝑊) |
hashdmpropge2.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
hashdmpropge2.d | ⊢ (𝜑 → 𝐷 ∈ 𝑌) |
hashdmpropge2.f | ⊢ (𝜑 → 𝐹 ∈ 𝑍) |
hashdmpropge2.n | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
hashdmpropge2.s | ⊢ (𝜑 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ 𝐹) |
Ref | Expression |
---|---|
hashdmpropge2 | ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashdmpropge2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑍) | |
2 | 1 | dmexd 7596 | . 2 ⊢ (𝜑 → dom 𝐹 ∈ V) |
3 | hashdmpropge2.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
4 | hashdmpropge2.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
5 | dmpropg 6039 | . . . . 5 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) | |
6 | 3, 4, 5 | syl2anc 587 | . . . 4 ⊢ (𝜑 → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) |
7 | hashdmpropge2.s | . . . . 5 ⊢ (𝜑 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ 𝐹) | |
8 | dmss 5735 | . . . . 5 ⊢ ({〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ 𝐹 → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ dom 𝐹) | |
9 | 7, 8 | syl 17 | . . . 4 ⊢ (𝜑 → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ dom 𝐹) |
10 | 6, 9 | eqsstrrd 3954 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ dom 𝐹) |
11 | hashdmpropge2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
12 | hashdmpropge2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
13 | prssg 4712 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) ↔ {𝐴, 𝐵} ⊆ dom 𝐹)) | |
14 | 11, 12, 13 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) ↔ {𝐴, 𝐵} ⊆ dom 𝐹)) |
15 | hashdmpropge2.n | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
16 | neeq1 3049 | . . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏)) | |
17 | neeq2 3050 | . . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵)) | |
18 | 16, 17 | rspc2ev 3583 | . . . . . . 7 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹 ∧ 𝐴 ≠ 𝐵) → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏) |
19 | 18 | 3expa 1115 | . . . . . 6 ⊢ (((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) ∧ 𝐴 ≠ 𝐵) → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏) |
20 | 19 | expcom 417 | . . . . 5 ⊢ (𝐴 ≠ 𝐵 → ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏)) |
21 | 15, 20 | syl 17 | . . . 4 ⊢ (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏)) |
22 | 14, 21 | sylbird 263 | . . 3 ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ dom 𝐹 → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏)) |
23 | 10, 22 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏) |
24 | hashge2el2difr 13835 | . 2 ⊢ ((dom 𝐹 ∈ V ∧ ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏) → 2 ≤ (♯‘dom 𝐹)) | |
25 | 2, 23, 24 | syl2anc 587 | 1 ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 Vcvv 3441 ⊆ wss 3881 {cpr 4527 〈cop 4531 class class class wbr 5030 dom cdm 5519 ‘cfv 6324 ≤ cle 10665 2c2 11680 ♯chash 13686 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-dju 9314 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-xnn0 11956 df-z 11970 df-uz 12232 df-fz 12886 df-hash 13687 |
This theorem is referenced by: structvtxvallem 26813 structgrssvtxlem 26816 |
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