Theorem snnen2o 9271

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 5371, ax-un 7754. (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.

Step	Hyp	Ref	Expression
1		df2o3 8513	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5313	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8515	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8525	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 2993	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4865	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1460	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 3009	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2940	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1797	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8521	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7302	. . . . . 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 6861	. . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴})
16		f1of1 6848	. . . . 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 1797	. 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o
20		snex 5442	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8516	. . 3 ⊢ 2o ∈ V
22		breng 8993	. . 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

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478  ∅c0 4339  {csn 4631  {cpr 4633   class class class wbr 5148  ^◡ccnv 5688  –1-1→wf1 6560  –1-1-onto→wf1o 6562  1_oc1o 8498  2_oc2o 8499   ≈ cen 8981

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1o 8505  df-2o 8506  df-en 8985

This theorem is referenced by:  1sdom2  9274  1sdom2dom  9281  pr2ne  10042  pmtrsn  19552  trivnsimpgd  20132