Theorem snnen2o 9274

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 5334, ax-un 7783. (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 8522	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5267	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8524	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8533	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 3013	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4820	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1491	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 3029	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2961	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1833	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8529	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7321	. . . . . 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 6867	. . . . 5 ⊢ (𝑓:{𝐴}⟶1-1onto2o → ◡𝑓:2o⟶1-1onto{𝐴})
16		f1of1 6853	. . . . 5 ⊢ (◡𝑓:2o⟶1-1onto{𝐴} → ◡𝑓:2o⟶1-1{𝐴})
17	15, 16	syl 18	. . . 4 ⊢ (𝑓:{𝐴}⟶1-1onto2o → ◡𝑓:2o⟶1-1{𝐴})
18	14, 17	mto 200	. . 3 ⊢ ¬ 𝑓:{𝐴}⟶1-1onto2o
19	18	nex 1833	. 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}⟶1-1onto2o
20		snex 5404	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8526	. . 3 ⊢ 2o ∈ V
22		breng 9020	. . 3 ⊢ (({𝐴} ∈ V ∧ 2o ∈ V) → ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}⟶1-1onto2o))
23	20, 21, 22	mp2an 705	. 2 ⊢ ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}⟶1-1onto2o)
24	19, 23	mtbir 326	1 ⊢ ¬ {𝐴} ≈ 2o

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  {csn 4588  {cpr 4590   class class class wbr 5108  ^◡ccnv 5657  ⟶1-1wf1 6563  ⟶1-1ontowf1o 6565  1_oc1o 8507  2_oc2o 8508   ≈ cen 9008

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-1o 8514  df-2o 8515  df-en 9012

This theorem is used by:  1sdom2  9277  1sdom2dom  9283  pr2ne  10123  pmtrsn  19784  trivnsimpgd  20364