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Mirrors > Home > MPE Home > Th. List > qfto | Structured version Visualization version GIF version |
Description: A quantifier-free way of expressing the total order predicate. (Contributed by Mario Carneiro, 22-Nov-2013.) |
Ref | Expression |
---|---|
qfto | ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxp 5593 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) | |
2 | brun 5119 | . . . . 5 ⊢ (𝑥(𝑅 ∪ ◡𝑅)𝑦 ↔ (𝑥𝑅𝑦 ∨ 𝑥◡𝑅𝑦)) | |
3 | df-br 5069 | . . . . 5 ⊢ (𝑥(𝑅 ∪ ◡𝑅)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) | |
4 | vex 3499 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
5 | vex 3499 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
6 | 4, 5 | brcnv 5755 | . . . . . 6 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥) |
7 | 6 | orbi2i 909 | . . . . 5 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥◡𝑅𝑦) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
8 | 2, 3, 7 | 3bitr3i 303 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅) ↔ (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
9 | 1, 8 | imbi12i 353 | . . 3 ⊢ ((〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) |
10 | 9 | 2albii 1821 | . 2 ⊢ (∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) |
11 | relxp 5575 | . . 3 ⊢ Rel (𝐴 × 𝐵) | |
12 | ssrel 5659 | . . 3 ⊢ (Rel (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅)))) | |
13 | 11, 12 | ax-mp 5 | . 2 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ (𝐴 × 𝐵) → 〈𝑥, 𝑦〉 ∈ (𝑅 ∪ ◡𝑅))) |
14 | r2al 3203 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥))) | |
15 | 10, 13, 14 | 3bitr4i 305 | 1 ⊢ ((𝐴 × 𝐵) ⊆ (𝑅 ∪ ◡𝑅) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∀wal 1535 ∈ wcel 2114 ∀wral 3140 ∪ cun 3936 ⊆ wss 3938 〈cop 4575 class class class wbr 5068 × cxp 5555 ◡ccnv 5556 Rel wrel 5562 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 df-cnv 5565 |
This theorem is referenced by: istsr2 17830 letsr 17839 |
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