Theorem snnen2o 9264

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 5361, ax-un 7738. (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 8496	. . . . . . . 8 ⊢ 2o = {∅, 1o}
2		0ex 5304	. . . . . . . . 9 ⊢ ∅ ∈ V
3		1oex 8498	. . . . . . . . 9 ⊢ 1o ∈ V
4		1n0 8510	. . . . . . . . . 10 ⊢ 1o ≠ ∅
5	4	necomi 2985	. . . . . . . . 9 ⊢ ∅ ≠ 1o
6		prnesn 4858	. . . . . . . . 9 ⊢ ((∅ ∈ V ∧ 1o ∈ V ∧ ∅ ≠ 1o) → {∅, 1o} ≠ {𝑥})
7	2, 3, 5, 6	mp3an 1458	. . . . . . . 8 ⊢ {∅, 1o} ≠ {𝑥}
8	1, 7	eqnetri 3001	. . . . . . 7 ⊢ 2o ≠ {𝑥}
9	8	neii 2932	. . . . . 6 ⊢ ¬ 2o = {𝑥}
10	9	nex 1795	. . . . 5 ⊢ ¬ ∃𝑥2o = {𝑥}
11		2on0 8504	. . . . . 6 ⊢ 2o ≠ ∅
12		f1cdmsn 7288	. . . . . 6 ⊢ ((◡𝑓:2o–1-1→{𝐴} ∧ 2o ≠ ∅) → ∃𝑥2o = {𝑥})
13	11, 12	mpan2 689	. . . . 5 ⊢ (◡𝑓:2o–1-1→{𝐴} → ∃𝑥2o = {𝑥})
14	10, 13	mto 196	. . . 4 ⊢ ¬ ◡𝑓:2o–1-1→{𝐴}
15		f1ocnv 6847	. . . . 5 ⊢ (𝑓:{𝐴}–1-1-onto→2o → ◡𝑓:2o–1-1-onto→{𝐴})
16		f1of1 6834	. . . . 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 196	. . 3 ⊢ ¬ 𝑓:{𝐴}–1-1-onto→2o
19	18	nex 1795	. 2 ⊢ ¬ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o
20		snex 5429	. . 3 ⊢ {𝐴} ∈ V
21		2oex 8499	. . 3 ⊢ 2o ∈ V
22		breng 8975	. . 3 ⊢ (({𝐴} ∈ V ∧ 2o ∈ V) → ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o))
23	20, 21, 22	mp2an 690	. 2 ⊢ ({𝐴} ≈ 2o ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→2o)
24	19, 23	mtbir 322	1 ⊢ ¬ {𝐴} ≈ 2o

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2930  Vcvv 3462  ∅c0 4322  {csn 4623  {cpr 4625   class class class wbr 5145  ^◡ccnv 5673  –1-1→wf1 6543  –1-1-onto→wf1o 6545  1_oc1o 8481  2_oc2o 8482   ≈ cen 8963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-1o 8488  df-2o 8489  df-en 8967

This theorem is referenced by:  1sdom2  9267  1sdom2dom  9274  pr2ne  10040  pmtrsn  19513  trivnsimpgd  20093