Theorem snnen2o 9250

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 5340, ax-un 7734. (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 8493	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5282	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8495	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8505	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 2987	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4841	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1463	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 3003	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2935	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1800	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8501	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7280	. . . . . 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 6835	. . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴})
16		f1of1 6822	. . . . 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 1800	. 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o
20		snex 5411	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8496	. . 3 ⊢ 2o ∈ V
22		breng 8973	. . 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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  Vcvv 3464  ∅c0 4313  {csn 4606  {cpr 4608   class class class wbr 5124  ^◡ccnv 5658  –1-1→wf1 6533  –1-1-onto→wf1o 6535  1_oc1o 8478  2_oc2o 8479   ≈ cen 8961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1o 8485  df-2o 8486  df-en 8965

This theorem is referenced by:  1sdom2  9253  1sdom2dom  9260  pr2ne  10023  pmtrsn  19505  trivnsimpgd  20085