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Theorem snnen2o 8190
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 7764 . . . 4 1𝑜 ∈ ω
2 php5 8189 . . . 4 (1𝑜 ∈ ω → ¬ 1𝑜 ≈ suc 1𝑜)
31, 2ax-mp 5 . . 3 ¬ 1𝑜 ≈ suc 1𝑜
4 ensn1g 8062 . . 3 (𝐴 ∈ V → {𝐴} ≈ 1𝑜)
5 df-2o 7606 . . . . . 6 2𝑜 = suc 1𝑜
65eqcomi 2660 . . . . 5 suc 1𝑜 = 2𝑜
76breq2i 4693 . . . 4 (1𝑜 ≈ suc 1𝑜 ↔ 1𝑜 ≈ 2𝑜)
8 ensymb 8045 . . . . . 6 ({𝐴} ≈ 1𝑜 ↔ 1𝑜 ≈ {𝐴})
9 entr 8049 . . . . . . 7 ((1𝑜 ≈ {𝐴} ∧ {𝐴} ≈ 2𝑜) → 1𝑜 ≈ 2𝑜)
109ex 449 . . . . . 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 319 . . 3 (¬ 1𝑜 ≈ suc 1𝑜 → ({𝐴} ≈ 1𝑜 → ¬ {𝐴} ≈ 2𝑜))
143, 4, 13mpsyl 68 . 2 (𝐴 ∈ V → ¬ {𝐴} ≈ 2𝑜)
15 2on0 7614 . . . 4 2𝑜 ≠ ∅
16 ensymb 8045 . . . . 5 (∅ ≈ 2𝑜 ↔ 2𝑜 ≈ ∅)
17 en0 8060 . . . . 5 (2𝑜 ≈ ∅ ↔ 2𝑜 = ∅)
1816, 17bitri 264 . . . 4 (∅ ≈ 2𝑜 ↔ 2𝑜 = ∅)
1915, 18nemtbir 2918 . . 3 ¬ ∅ ≈ 2𝑜
20 snprc 4285 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
2120biimpi 206 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
2221breq1d 4695 . . 3 𝐴 ∈ V → ({𝐴} ≈ 2𝑜 ↔ ∅ ≈ 2𝑜))
2319, 22mtbiri 316 . 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 1523  wcel 2030  Vcvv 3231  c0 3948  {csn 4210   class class class wbr 4685  suc csuc 5763  ωcom 7107  1𝑜c1o 7598  2𝑜c2o 7599  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-2o 7606  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000
This theorem is referenced by:  pmtrsn  17985
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