Theorem snnen2o 9129

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 5301, ax-un 7668. (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 8393	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5243	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8395	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8403	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 2982	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4809	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1463	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 2998	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2930	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1801	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8399	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7216	. . . . . 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 6775	. . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴})
16		f1of1 6762	. . . . 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 1801	. 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o
20		snex 5372	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8396	. . 3 ⊢ 2o ∈ V
22		breng 8878	. . 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 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436  ∅c0 4280  {csn 4573  {cpr 4575   class class class wbr 5089  ^◡ccnv 5613  –1-1→wf1 6478  –1-1-onto→wf1o 6480  1_oc1o 8378  2_oc2o 8379   ≈ cen 8866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-1o 8385  df-2o 8386  df-en 8870

This theorem is referenced by:  1sdom2  9132  1sdom2dom  9138  pr2ne  9896  pmtrsn  19431  trivnsimpgd  20011