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Mirrors > Home > MPE Home > Th. List > caovord3 | Structured version Visualization version GIF version |
Description: Ordering law. (Contributed by NM, 29-Feb-1996.) |
Ref | Expression |
---|---|
caovord.1 | ⊢ 𝐴 ∈ V |
caovord.2 | ⊢ 𝐵 ∈ V |
caovord.3 | ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) |
caovord2.3 | ⊢ 𝐶 ∈ V |
caovord2.com | ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) |
caovord3.4 | ⊢ 𝐷 ∈ V |
Ref | Expression |
---|---|
caovord3 | ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caovord.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | caovord2.3 | . . . . 5 ⊢ 𝐶 ∈ V | |
3 | caovord.3 | . . . . 5 ⊢ (𝑧 ∈ 𝑆 → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
4 | caovord.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
5 | caovord2.com | . . . . 5 ⊢ (𝑥𝐹𝑦) = (𝑦𝐹𝑥) | |
6 | 1, 2, 3, 4, 5 | caovord2 7602 | . . . 4 ⊢ (𝐵 ∈ 𝑆 → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵))) |
7 | 6 | adantr 481 | . . 3 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵))) |
8 | breq1 5144 | . . 3 ⊢ ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) | |
9 | 7, 8 | sylan9bb 510 | . 2 ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) |
10 | caovord3.4 | . . . 4 ⊢ 𝐷 ∈ V | |
11 | 10, 4, 3 | caovord 7601 | . . 3 ⊢ (𝐶 ∈ 𝑆 → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) |
12 | 11 | ad2antlr 725 | . 2 ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) |
13 | 9, 12 | bitr4d 281 | 1 ⊢ (((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ (𝐴𝐹𝐵) = (𝐶𝐹𝐷)) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3473 class class class wbr 5141 (class class class)co 7393 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-iota 6484 df-fv 6540 df-ov 7396 |
This theorem is referenced by: genpnnp 10982 ltsrpr 11054 |
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