Theorem snnen2o 9215

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 5330, ax-un 7736. (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 8463	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5264	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8465	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8474	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 3009	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4820	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1490	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 3025	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2957	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1833	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8470	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7283	. . . . . 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 6830	. . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴})
16		f1of1 6816	. . . . 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 5404	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8467	. . 3 ⊢ 2o ∈ V
22		breng 8961	. . 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 2145   ≠ wne 2955  Vcvv 3450  ∅c0 4279  {csn 4584  {cpr 4586   class class class wbr 5103  ^◡ccnv 5654  –1-1→wf1 6530  –1-1-onto→wf1o 6532  1_oc1o 8448  2_oc2o 8449   ≈ cen 8949

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-1o 8455  df-2o 8456  df-en 8953

This theorem is used by:  1sdom2  9218  1sdom2dom  9224  pr2ne  10008  pmtrsn  19646  trivnsimpgd  20226