Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > enqex | Structured version Visualization version GIF version |
Description: The equivalence relation for positive fractions exists. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.) |
Ref | Expression |
---|---|
enqex | ⊢ ~Q ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | niex 9703 | . . . 4 ⊢ N ∈ V | |
2 | 1, 1 | xpex 6962 | . . 3 ⊢ (N × N) ∈ V |
3 | 2, 2 | xpex 6962 | . 2 ⊢ ((N × N) × (N × N)) ∈ V |
4 | df-enq 9733 | . . 3 ⊢ ~Q = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} | |
5 | opabssxp 5193 | . . 3 ⊢ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ (N × N) ∧ 𝑦 ∈ (N × N)) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 〈𝑧, 𝑤〉 ∧ 𝑦 = 〈𝑣, 𝑢〉) ∧ (𝑧 ·N 𝑢) = (𝑤 ·N 𝑣)))} ⊆ ((N × N) × (N × N)) | |
6 | 4, 5 | eqsstri 3635 | . 2 ⊢ ~Q ⊆ ((N × N) × (N × N)) |
7 | 3, 6 | ssexi 4803 | 1 ⊢ ~Q ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 〈cop 4183 {copab 4712 × cxp 5112 (class class class)co 6650 Ncnpi 9666 ·N cmi 9668 ~Q ceq 9673 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 df-ni 9694 df-enq 9733 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |