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| Mirrors > Home > MPE Home > Th. List > tpprceq3 | Structured version Visualization version GIF version | ||
| Description: An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.) |
| Ref | Expression |
|---|---|
| tpprceq3 | ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ianor 984 | . 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵)) | |
| 2 | prprc2 4766 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵}) | |
| 3 | 2 | uneq1d 4167 | . . . 4 ⊢ (¬ 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴})) |
| 4 | tprot 4749 | . . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
| 5 | df-tp 4631 | . . . . 5 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴}) | |
| 6 | 4, 5 | eqtri 2765 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴}) |
| 7 | prcom 4732 | . . . . 5 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
| 8 | df-pr 4629 | . . . . 5 ⊢ {𝐵, 𝐴} = ({𝐵} ∪ {𝐴}) | |
| 9 | 7, 8 | eqtri 2765 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴}) |
| 10 | 3, 6, 9 | 3eqtr4g 2802 | . . 3 ⊢ (¬ 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
| 11 | nne 2944 | . . . 4 ⊢ (¬ 𝐶 ≠ 𝐵 ↔ 𝐶 = 𝐵) | |
| 12 | tppreq3 4759 | . . . . 5 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | |
| 13 | 12 | eqcoms 2745 | . . . 4 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
| 14 | 11, 13 | sylbi 217 | . . 3 ⊢ (¬ 𝐶 ≠ 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
| 15 | 10, 14 | jaoi 858 | . 2 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
| 16 | 1, 15 | sylbi 217 | 1 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∪ cun 3949 {csn 4626 {cpr 4628 {ctp 4630 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-nul 4334 df-sn 4627 df-pr 4629 df-tp 4631 |
| This theorem is referenced by: tppreqb 4805 1to3vfriswmgr 30299 |
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