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Theorem snnen2o 8093
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.)
Assertion
Ref Expression
snnen2o ¬ {𝐴} ≈ 2𝑜

Proof of Theorem snnen2o
StepHypRef Expression
1 1onn 7664 . . . 4 1𝑜 ∈ ω
2 php5 8092 . . . 4 (1𝑜 ∈ ω → ¬ 1𝑜 ≈ suc 1𝑜)
31, 2ax-mp 5 . . 3 ¬ 1𝑜 ≈ suc 1𝑜
4 ensn1g 7965 . . 3 (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
5 df-2o 7506 . . . . . 6 2𝑜 = suc 1𝑜
65eqcomi 2630 . . . . 5 suc 1𝑜 = 2𝑜
76breq2i 4621 . . . 4 (1𝑜 ≈ suc 1𝑜 ↔ 1𝑜 ≈ 2𝑜)
8 ensymb 7948 . . . . . 6 ({𝐴} ≈ 1𝑜 ↔ 1𝑜 ≈ {𝐴})
9 entr 7952 . . . . . . 7 ((1𝑜 ≈ {𝐴} ∧ {𝐴} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
109ex 450 . . . . . 6 (1𝑜 ≈ {𝐴} → ({𝐴} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
118, 10sylbi 207 . . . . 5 ({𝐴} ≈ 1𝑜 → ({𝐴} ≈ 2𝑜 → 1𝑜 ≈ 2𝑜))
1211con3rr3 151 . . . 4 (¬ 1𝑜 ≈ 2𝑜 → ({𝐴} ≈ 1𝑜 → ¬ {𝐴} ≈ 2𝑜))
137, 12sylnbi 320 . . 3 (¬ 1𝑜 ≈ suc 1𝑜 → ({𝐴} ≈ 1𝑜 → ¬ {𝐴} ≈ 2𝑜))
143, 4, 13mpsyl 68 . 2 (𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
15 2on0 7514 . . . 4 2𝑜 ≠ ∅
16 ensymb 7948 . . . . 5 (∅ ≈ 2𝑜 ↔ 2𝑜 ≈ ∅)
17 en0 7963 . . . . 5 (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅)
1816, 17bitri 264 . . . 4 (∅ ≈ 2𝑜 ↔ 2𝑜 = ∅)
1915, 18nemtbir 2885 . . 3 ¬ ∅ ≈ 2𝑜
20 snprc 4223 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
2120biimpi 206 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
2221breq1d 4623 . . 3 𝐴 ∈ V → ({𝐴} ≈ 2𝑜 ↔ ∅ ≈ 2𝑜))
2319, 22mtbiri 317 . 2 𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
2414, 23pm2.61i 176 1 ¬ {𝐴} ≈ 2𝑜
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  {csn 4148   class class class wbr 4613  suc csuc 5684  ωcom 7012  1𝑜c1o 7498  2𝑜c2o 7499  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-1o 7505  df-2o 7506  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902
This theorem is referenced by:  pmtrsn  17860
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