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Theorem snnen2o 9264
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 5361, ax-un 7738. (Revised by BTernaryTau, 1-Dec-2024.)
Assertion
Ref Expression
snnen2o ¬ {𝐴} ≈ 2o

Proof of Theorem snnen2o
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 df2o3 8496 . . . . . . . 8 2o = {∅, 1o}
2 0ex 5304 . . . . . . . . 9 ∅ ∈ V
3 1oex 8498 . . . . . . . . 9 1o ∈ V
4 1n0 8510 . . . . . . . . . 10 1o ≠ ∅
54necomi 2985 . . . . . . . . 9 ∅ ≠ 1o
6 prnesn 4858 . . . . . . . . 9 ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
72, 3, 5, 6mp3an 1458 . . . . . . . 8 {∅, 1o} ≠ {𝑥}
81, 7eqnetri 3001 . . . . . . 7 2o ≠ {𝑥}
98neii 2932 . . . . . 6 ¬ 2o = {𝑥}
109nex 1795 . . . . 5 ¬ ∃𝑥2o = {𝑥}
11 2on0 8504 . . . . . 6 2o ≠ ∅
12 f1cdmsn 7288 . . . . . 6 ((𝑓:2o1-1→{𝐴} ∧ 2o ≠ ∅) → ∃𝑥2o = {𝑥})
1311, 12mpan2 689 . . . . 5 (𝑓:2o1-1→{𝐴} → ∃𝑥2o = {𝑥})
1410, 13mto 196 . . . 4 ¬ 𝑓:2o1-1→{𝐴}
15 f1ocnv 6847 . . . . 5 (𝑓:{𝐴}–1-1-onto→2o𝑓:2o1-1-onto→{𝐴})
16 f1of1 6834 . . . . 5 (𝑓:2o1-1-onto→{𝐴} → 𝑓:2o1-1→{𝐴})
1715, 16syl 17 . . . 4 (𝑓:{𝐴}–1-1-onto→2o𝑓:2o1-1→{𝐴})
1814, 17mto 196 . . 3 ¬ 𝑓:{𝐴}–1-1-onto→2o
1918nex 1795 . 2 ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o
20 snex 5429 . . 3 {𝐴} ∈ V
21 2oex 8499 . . 3 2o ∈ V
22 breng 8975 . . 3 (({𝐴} ∈ V ∧ 2o ∈ V) → ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o))
2320, 21, 22mp2an 690 . 2 ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o)
2419, 23mtbir 322 1 ¬ {𝐴} ≈ 2o
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205   = wceq 1534  wex 1774  wcel 2099  wne 2930  Vcvv 3462  c0 4322  {csn 4623  {cpr 4625   class class class wbr 5145  ccnv 5673  1-1wf1 6543  1-1-ontowf1o 6545  1oc1o 8481  2oc2o 8482  cen 8963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-1o 8488  df-2o 8489  df-en 8967
This theorem is referenced by:  1sdom2  9267  1sdom2dom  9274  pr2ne  10040  pmtrsn  19513  trivnsimpgd  20093
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