|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > snnen2o | Structured version Visualization version GIF version | ||
| Description: A singleton {𝐴} is never equinumerous with the ordinal number 2. This holds for proper singletons (𝐴 ∈ V) as well as for singletons being the empty set (𝐴 ∉ V). (Contributed by AV, 6-Aug-2019.) Avoid ax-pow 5364, ax-un 7756. (Revised by BTernaryTau, 1-Dec-2024.) | 
| Ref | Expression | 
|---|---|
| snnen2o | ⊢ ¬ {𝐴} ≈ 2o | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df2o3 8515 | . . . . . . . 8 ⊢ 2o = {∅, 1o} | |
| 2 | 0ex 5306 | . . . . . . . . 9 ⊢ ∅ ∈ V | |
| 3 | 1oex 8517 | . . . . . . . . 9 ⊢ 1o ∈ V | |
| 4 | 1n0 8527 | . . . . . . . . . 10 ⊢ 1o ≠ ∅ | |
| 5 | 4 | necomi 2994 | . . . . . . . . 9 ⊢ ∅ ≠ 1o | 
| 6 | prnesn 4859 | . . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥}) | |
| 7 | 2, 3, 5, 6 | mp3an 1462 | . . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥} | 
| 8 | 1, 7 | eqnetri 3010 | . . . . . . 7 ⊢ 2o ≠ {𝑥} | 
| 9 | 8 | neii 2941 | . . . . . 6 ⊢ ¬ 2o = {𝑥} | 
| 10 | 9 | nex 1799 | . . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥} | 
| 11 | 2on0 8523 | . . . . . 6 ⊢ 2o ≠ ∅ | |
| 12 | f1cdmsn 7303 | . . . . . 6 ⊢ ((◡𝑓:2o–1-1→{𝐴} ∧ 2o ≠ ∅) → ∃𝑥2o = {𝑥}) | |
| 13 | 11, 12 | mpan2 691 | . . . . 5 ⊢ (◡𝑓:2o–1-1→{𝐴} → ∃𝑥2o = {𝑥}) | 
| 14 | 10, 13 | mto 197 | . . . 4 ⊢ ¬ ◡𝑓:2o–1-1→{𝐴} | 
| 15 | f1ocnv 6859 | . . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴}) | |
| 16 | f1of1 6846 | . . . . 5 ⊢ (◡𝑓:2o–1-1-onto→{𝐴} → ◡𝑓:2o–1-1→{𝐴}) | |
| 17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1→{𝐴}) | 
| 18 | 14, 17 | mto 197 | . . 3 ⊢ ¬ 𝑓:{𝐴}–1-1-onto→2o | 
| 19 | 18 | nex 1799 | . 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o | 
| 20 | snex 5435 | . . 3 ⊢ {𝐴} ∈ V | |
| 21 | 2oex 8518 | . . 3 ⊢ 2o ∈ V | |
| 22 | breng 8995 | . . 3 ⊢ (({𝐴} ∈ V ∧ 2o ∈ V) → ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o)) | |
| 23 | 20, 21, 22 | mp2an 692 | . 2 ⊢ ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o) | 
| 24 | 19, 23 | mtbir 323 | 1 ⊢ ¬ {𝐴} ≈ 2o | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1539 ∃wex 1778 ∈ wcel 2107 ≠ wne 2939 Vcvv 3479 ∅c0 4332 {csn 4625 {cpr 4627 class class class wbr 5142 ◡ccnv 5683 –1-1→wf1 6557 –1-1-onto→wf1o 6559 1oc1o 8500 2oc2o 8501 ≈ cen 8983 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-1o 8507 df-2o 8508 df-en 8987 | 
| This theorem is referenced by: 1sdom2 9277 1sdom2dom 9284 pr2ne 10045 pmtrsn 19538 trivnsimpgd 20118 | 
| Copyright terms: Public domain | W3C validator |