Theorem 1sdomOLD 9262

Description: Obsolete version of 1sdom 9261 as of 30-Dec-2024. (Contributed by Mario Carneiro, 12-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
1sdomOLD	⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem 1sdomOLD

Dummy variables 𝑓 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5128	. 2 ⊢ (𝑎 = 𝐴 → (1o ≺ 𝑎 ↔ 1o ≺ 𝐴))
2		rexeq 3305	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
3	2	rexeqbi1dv 3322	. 2 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
4		1onn 8657	. . . 4 ⊢ 1o ∈ ω
5		sucdom 9248	. . . 4 ⊢ (1o ∈ ω → (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎))
6	4, 5	ax-mp 5	. . 3 ⊢ (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎)
7		df-2o 8486	. . . 4 ⊢ 2o = suc 1o
8	7	breq1i 5131	. . 3 ⊢ (2o ≼ 𝑎 ↔ suc 1o ≼ 𝑎)
9		2dom 9049	. . . 4 ⊢ (2o ≼ 𝑎 → ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
10		df2o3 8493	. . . . 5 ⊢ 2o = {∅, 1o}
11		vex 3468	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		vex 3468	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
13		0ex 5282	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14		1oex 8495	. . . . . . . . . . . 12 ⊢ 1o ∈ V
15	11, 12, 13, 14	funpr 6597	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
16		df-ne 2934	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
17		1n0 8505	. . . . . . . . . . . . . . 15 ⊢ 1o ≠ ∅
18	17	necomi 2987	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1o
19	13, 14, 11, 12	fpr 7149	. . . . . . . . . . . . . 14 ⊢ (∅ ≠ 1o → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦})
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦}
21		df-f1 6541	. . . . . . . . . . . . 13 ⊢ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ↔ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦} ∧ Fun ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
22	20, 21	mpbiran 709	. . . . . . . . . . . 12 ⊢ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ↔ Fun ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
23	13, 11	cnvsn 6220	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
24	14, 12	cnvsn 6220	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨1o, 𝑦⟩} = {⟨𝑦, 1o⟩}
25	23, 24	uneq12i 4146	. . . . . . . . . . . . . 14 ⊢ (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
26		df-pr 4609	. . . . . . . . . . . . . . . 16 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
27	26	cnveqi 5859	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
28		cnvun 6136	. . . . . . . . . . . . . . 15 ⊢ ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
29	27, 28	eqtri 2759	. . . . . . . . . . . . . 14 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
30		df-pr 4609	. . . . . . . . . . . . . 14 ⊢ {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
31	25, 29, 30	3eqtr4i 2769	. . . . . . . . . . . . 13 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩}
32	31	funeqi 6562	. . . . . . . . . . . 12 ⊢ (Fun ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ↔ Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
33	22, 32	bitr2i 276	. . . . . . . . . . 11 ⊢ (Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
34	15, 16, 33	3imtr3i 291	. . . . . . . . . 10 ⊢ (¬ 𝑥 = 𝑦 → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
35		prssi 4802	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36		f1ss 6784	. . . . . . . . . 10 ⊢ (({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎)
37	34, 35, 36	syl2an 596	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎)
38		prex 5412	. . . . . . . . . 10 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ V
39		f1eq1 6774	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} → (𝑓:{∅, 1o}–1-1→𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎))
40	38, 39	spcev 3590	. . . . . . . . 9 ⊢ ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎 → ∃𝑓 𝑓:{∅, 1o}–1-1→𝑎)
41	37, 40	syl 17	. . . . . . . 8 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → ∃𝑓 𝑓:{∅, 1o}–1-1→𝑎)
42		vex 3468	. . . . . . . . 9 ⊢ 𝑎 ∈ V
43	42	brdom 8980	. . . . . . . 8 ⊢ ({∅, 1o} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1o}–1-1→𝑎)
44	41, 43	sylibr 234	. . . . . . 7 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {∅, 1o} ≼ 𝑎)
45	44	expcom 413	. . . . . 6 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎))
46	45	rexlimivv 3187	. . . . 5 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎)
47	10, 46	eqbrtrid 5159	. . . 4 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → 2o ≼ 𝑎)
48	9, 47	impbii 209	. . 3 ⊢ (2o ≼ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
49	6, 8, 48	3bitr2i 299	. 2 ⊢ (1o ≺ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
50	1, 3, 49	vtoclbg 3541	1 ⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∃wrex 3061   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  {csn 4606  {cpr 4608  ⟨cop 4612   class class class wbr 5124  ^◡ccnv 5658  suc csuc 6359  Fun wfun 6530  ⟶wf 6532  –1-1→wf1 6533  ωcom 7866  1_oc1o 8478  2_oc2o 8479   ≼ cdom 8962   ≺ csdm 8963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7867  df-1o 8485  df-2o 8486  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968

This theorem is referenced by: (None)