Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 978 | . 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵)) | |
2 | prprc2 4704 | . . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵}) | |
3 | 2 | uneq1d 4140 | . . . 4 ⊢ (¬ 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴})) |
4 | tprot 4687 | . . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴} | |
5 | df-tp 4574 | . . . . 5 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴}) | |
6 | 4, 5 | eqtri 2846 | . . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴}) |
7 | prcom 4670 | . . . . 5 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴} | |
8 | df-pr 4572 | . . . . 5 ⊢ {𝐵, 𝐴} = ({𝐵} ∪ {𝐴}) | |
9 | 7, 8 | eqtri 2846 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴}) |
10 | 3, 6, 9 | 3eqtr4g 2883 | . . 3 ⊢ (¬ 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
11 | nne 3022 | . . . 4 ⊢ (¬ 𝐶 ≠ 𝐵 ↔ 𝐶 = 𝐵) | |
12 | tppreq3 4697 | . . . . 5 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) | |
13 | 12 | eqcoms 2831 | . . . 4 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
14 | 11, 13 | sylbi 219 | . . 3 ⊢ (¬ 𝐶 ≠ 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
15 | 10, 14 | jaoi 853 | . 2 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
16 | 1, 15 | sylbi 219 | 1 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∪ cun 3936 {csn 4569 {cpr 4571 {ctp 4573 |
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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-v 3498 df-dif 3941 df-un 3943 df-nul 4294 df-sn 4570 df-pr 4572 df-tp 4574 |
This theorem is referenced by: tppreqb 4740 1to3vfriswmgr 28061 |
Copyright terms: Public domain | W3C validator |