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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.) |
Ref | Expression |
---|---|
snnen2o | ⊢ ¬ {𝐴} ≈ 2o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 7985 | . . . 4 ⊢ 1o ∈ ω | |
2 | php5 8416 | . . . 4 ⊢ (1o ∈ ω → ¬ 1o ≈ suc 1o) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ¬ 1o ≈ suc 1o |
4 | ensn1g 8286 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
5 | df-2o 7826 | . . . . . 6 ⊢ 2o = suc 1o | |
6 | 5 | eqcomi 2833 | . . . . 5 ⊢ suc 1o = 2o |
7 | 6 | breq2i 4880 | . . . 4 ⊢ (1o ≈ suc 1o ↔ 1o ≈ 2o) |
8 | ensymb 8269 | . . . . . 6 ⊢ ({𝐴} ≈ 1o ↔ 1o ≈ {𝐴}) | |
9 | entr 8273 | . . . . . . 7 ⊢ ((1o ≈ {𝐴} ∧ {𝐴} ≈ 2o) → 1o ≈ 2o) | |
10 | 9 | ex 403 | . . . . . 6 ⊢ (1o ≈ {𝐴} → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
11 | 8, 10 | sylbi 209 | . . . . 5 ⊢ ({𝐴} ≈ 1o → ({𝐴} ≈ 2o → 1o ≈ 2o)) |
12 | 11 | con3rr3 153 | . . . 4 ⊢ (¬ 1o ≈ 2o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
13 | 7, 12 | sylnbi 322 | . . 3 ⊢ (¬ 1o ≈ suc 1o → ({𝐴} ≈ 1o → ¬ {𝐴} ≈ 2o)) |
14 | 3, 4, 13 | mpsyl 68 | . 2 ⊢ (𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
15 | 2on0 7835 | . . . 4 ⊢ 2o ≠ ∅ | |
16 | ensymb 8269 | . . . . 5 ⊢ (∅ ≈ 2o ↔ 2o ≈ ∅) | |
17 | en0 8284 | . . . . 5 ⊢ (2o ≈ ∅ ↔ 2o = ∅) | |
18 | 16, 17 | bitri 267 | . . . 4 ⊢ (∅ ≈ 2o ↔ 2o = ∅) |
19 | 15, 18 | nemtbir 3093 | . . 3 ⊢ ¬ ∅ ≈ 2o |
20 | snprc 4470 | . . . . 5 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
21 | 20 | biimpi 208 | . . . 4 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅) |
22 | 21 | breq1d 4882 | . . 3 ⊢ (¬ 𝐴 ∈ V → ({𝐴} ≈ 2o ↔ ∅ ≈ 2o)) |
23 | 19, 22 | mtbiri 319 | . 2 ⊢ (¬ 𝐴 ∈ V → ¬ {𝐴} ≈ 2o) |
24 | 14, 23 | pm2.61i 177 | 1 ⊢ ¬ {𝐴} ≈ 2o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1658 ∈ wcel 2166 Vcvv 3413 ∅c0 4143 {csn 4396 class class class wbr 4872 suc csuc 5964 ωcom 7325 1oc1o 7818 2oc2o 7819 ≈ cen 8218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-br 4873 df-opab 4935 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-om 7326 df-1o 7825 df-2o 7826 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 |
This theorem is referenced by: pmtrsn 18289 |
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