Theorem 1sdomOLD 9196

Description: Obsolete version of 1sdom 9195 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 5111	. 2 ⊢ (𝑎 = 𝐴 → (1o ≺ 𝑎 ↔ 1o ≺ 𝐴))
2		rexeq 3295	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
3	2	rexeqbi1dv 3312	. 2 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
4		1onn 8604	. . . 4 ⊢ 1o ∈ ω
5		sucdom 9183	. . . 4 ⊢ (1o ∈ ω → (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎))
6	4, 5	ax-mp 5	. . 3 ⊢ (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎)
7		df-2o 8435	. . . 4 ⊢ 2o = suc 1o
8	7	breq1i 5114	. . 3 ⊢ (2o ≼ 𝑎 ↔ suc 1o ≼ 𝑎)
9		2dom 9001	. . . 4 ⊢ (2o ≼ 𝑎 → ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
10		df2o3 8442	. . . . 5 ⊢ 2o = {∅, 1o}
11		vex 3451	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		vex 3451	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
13		0ex 5262	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14		1oex 8444	. . . . . . . . . . . 12 ⊢ 1o ∈ V
15	11, 12, 13, 14	funpr 6572	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
16		df-ne 2926	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
17		1n0 8452	. . . . . . . . . . . . . . 15 ⊢ 1o ≠ ∅
18	17	necomi 2979	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1o
19	13, 14, 11, 12	fpr 7126	. . . . . . . . . . . . . 14 ⊢ (∅ ≠ 1o → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦})
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦}
21		df-f1 6516	. . . . . . . . . . . . 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 6199	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
24	14, 12	cnvsn 6199	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨1o, 𝑦⟩} = {⟨𝑦, 1o⟩}
25	23, 24	uneq12i 4129	. . . . . . . . . . . . . 14 ⊢ (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
26		df-pr 4592	. . . . . . . . . . . . . . . 16 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
27	26	cnveqi 5838	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
28		cnvun 6115	. . . . . . . . . . . . . . 15 ⊢ ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
29	27, 28	eqtri 2752	. . . . . . . . . . . . . 14 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
30		df-pr 4592	. . . . . . . . . . . . . 14 ⊢ {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
31	25, 29, 30	3eqtr4i 2762	. . . . . . . . . . . . 13 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩}
32	31	funeqi 6537	. . . . . . . . . . . 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 4785	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36		f1ss 6761	. . . . . . . . . 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 5392	. . . . . . . . . 10 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ V
39		f1eq1 6751	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} → (𝑓:{∅, 1o}–1-1→𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎))
40	38, 39	spcev 3572	. . . . . . . . 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 3451	. . . . . . . . 9 ⊢ 𝑎 ∈ V
43	42	brdom 8932	. . . . . . . 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 3179	. . . . 5 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎)
47	10, 46	eqbrtrid 5142	. . . 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 3523	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 2925  ∃wrex 3053   ∪ cun 3912   ⊆ wss 3914  ∅c0 4296  {csn 4589  {cpr 4591  ⟨cop 4595   class class class wbr 5107  ^◡ccnv 5637  suc csuc 6334  Fun wfun 6505  ⟶wf 6507  –1-1→wf1 6508  ωcom 7842  1_oc1o 8427  2_oc2o 8428   ≼ cdom 8916   ≺ csdm 8917

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-2o 8435  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922

This theorem is referenced by: (None)