Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 7615 | . 2 ⊢ (𝜑 → dom 𝐹 ∈ V) |
3 | hashdmpropge2.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
4 | hashdmpropge2.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
5 | dmpropg 6072 | . . . . 5 ⊢ ((𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑌) → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) | |
6 | 3, 4, 5 | syl2anc 586 | . . . 4 ⊢ (𝜑 → dom {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} = {𝐴, 𝐵}) |
7 | hashdmpropge2.s | . . . . 5 ⊢ (𝜑 → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ⊆ 𝐹) | |
8 | dmss 5771 | . . . . 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 4006 | . . 3 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ dom 𝐹) |
11 | hashdmpropge2.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
12 | hashdmpropge2.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊) | |
13 | prssg 4752 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) ↔ {𝐴, 𝐵} ⊆ dom 𝐹)) | |
14 | 11, 12, 13 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) ↔ {𝐴, 𝐵} ⊆ dom 𝐹)) |
15 | hashdmpropge2.n | . . . . 5 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
16 | neeq1 3078 | . . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝑏 ↔ 𝐴 ≠ 𝑏)) | |
17 | neeq2 3079 | . . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝐴 ≠ 𝑏 ↔ 𝐴 ≠ 𝐵)) | |
18 | 16, 17 | rspc2ev 3635 | . . . . . . 7 ⊢ ((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹 ∧ 𝐴 ≠ 𝐵) → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏) |
19 | 18 | 3expa 1114 | . . . . . 6 ⊢ (((𝐴 ∈ dom 𝐹 ∧ 𝐵 ∈ dom 𝐹) ∧ 𝐴 ≠ 𝐵) → ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏) |
20 | 19 | expcom 416 | . . . . 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 262 | . . 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 13840 | . 2 ⊢ ((dom 𝐹 ∈ V ∧ ∃𝑎 ∈ dom 𝐹∃𝑏 ∈ dom 𝐹 𝑎 ≠ 𝑏) → 2 ≤ (♯‘dom 𝐹)) | |
25 | 2, 23, 24 | syl2anc 586 | 1 ⊢ (𝜑 → 2 ≤ (♯‘dom 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 {cpr 4569 〈cop 4573 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 ≤ cle 10676 2c2 11693 ♯chash 13691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-dju 9330 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-xnn0 11969 df-z 11983 df-uz 12245 df-fz 12894 df-hash 13692 |
This theorem is referenced by: structvtxvallem 26805 structgrssvtxlem 26808 |
Copyright terms: Public domain | W3C validator |