Theorem snnen2o 9212

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 5338, ax-un 7742. (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 8467	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5272	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8469	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8478	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 3014	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4827	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1490	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 3030	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2962	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1833	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8474	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7289	. . . . . 6 ⊢ ((◡𝑓:2o–1-1→{𝐴} ∧ 2o ≠ ∅) → ∃𝑥2o = {𝑥})
13	11, 12	mpan2 704	. . . . 5 ⊢ (◡𝑓:2o–1-1→{𝐴} → ∃𝑥2o = {𝑥})
14	10, 13	mto 200	. . . 4 ⊢ ¬ ◡𝑓:2o–1-1→{𝐴}
15		f1ocnv 6837	. . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴})
16		f1of1 6823	. . . . 5 ⊢ (◡𝑓:2o–1-1-onto→{𝐴} → ◡𝑓:2o–1-1→{𝐴})
17	15, 16	syl 18	. . . 4 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1→{𝐴})
18	14, 17	mto 200	. . 3 ⊢ ¬ 𝑓:{𝐴}–1-1-onto→2o
19	18	nex 1833	. 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o
20		snex 5412	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8471	. . 3 ⊢ 2o ∈ V
22		breng 8958	. . 3 ⊢ (({𝐴} ∈ V ∧ 2o ∈ V) → ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o))
23	20, 21, 22	mp2an 705	. 2 ⊢ ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o)
24	19, 23	mtbir 326	1 ⊢ ¬ {𝐴} ≈ 2o

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457  ∅c0 4286  {csn 4591  {cpr 4593   class class class wbr 5111  ^◡ccnv 5662  –1-1→wf1 6537  –1-1-onto→wf1o 6539  1_oc1o 8452  2_oc2o 8453   ≈ cen 8946

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1o 8459  df-2o 8460  df-en 8950

This theorem is used by:  1sdom2  9215  1sdom2dom  9221  pr2ne  10005  pmtrsn  19633  trivnsimpgd  20213