Theorem 1sdomOLD 9283

Description: Obsolete version of 1sdom 9282 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 5152	. 2 ⊢ (𝑎 = 𝐴 → (1o ≺ 𝑎 ↔ 1o ≺ 𝐴))
2		rexeq 3320	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
3	2	rexeqbi1dv 3337	. 2 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
4		1onn 8677	. . . 4 ⊢ 1o ∈ ω
5		sucdom 9269	. . . 4 ⊢ (1o ∈ ω → (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎))
6	4, 5	ax-mp 5	. . 3 ⊢ (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎)
7		df-2o 8506	. . . 4 ⊢ 2o = suc 1o
8	7	breq1i 5155	. . 3 ⊢ (2o ≼ 𝑎 ↔ suc 1o ≼ 𝑎)
9		2dom 9069	. . . 4 ⊢ (2o ≼ 𝑎 → ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
10		df2o3 8513	. . . . 5 ⊢ 2o = {∅, 1o}
11		vex 3482	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		vex 3482	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
13		0ex 5313	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14		1oex 8515	. . . . . . . . . . . 12 ⊢ 1o ∈ V
15	11, 12, 13, 14	funpr 6624	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
16		df-ne 2939	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
17		1n0 8525	. . . . . . . . . . . . . . 15 ⊢ 1o ≠ ∅
18	17	necomi 2993	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1o
19	13, 14, 11, 12	fpr 7174	. . . . . . . . . . . . . 14 ⊢ (∅ ≠ 1o → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦})
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦}
21		df-f1 6568	. . . . . . . . . . . . 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 6248	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
24	14, 12	cnvsn 6248	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨1o, 𝑦⟩} = {⟨𝑦, 1o⟩}
25	23, 24	uneq12i 4176	. . . . . . . . . . . . . 14 ⊢ (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
26		df-pr 4634	. . . . . . . . . . . . . . . 16 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
27	26	cnveqi 5888	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
28		cnvun 6165	. . . . . . . . . . . . . . 15 ⊢ ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
29	27, 28	eqtri 2763	. . . . . . . . . . . . . 14 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
30		df-pr 4634	. . . . . . . . . . . . . 14 ⊢ {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
31	25, 29, 30	3eqtr4i 2773	. . . . . . . . . . . . 13 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩}
32	31	funeqi 6589	. . . . . . . . . . . 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 4826	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36		f1ss 6810	. . . . . . . . . 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 5443	. . . . . . . . . 10 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ V
39		f1eq1 6800	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} → (𝑓:{∅, 1o}–1-1→𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎))
40	38, 39	spcev 3606	. . . . . . . . 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 3482	. . . . . . . . 9 ⊢ 𝑎 ∈ V
43	42	brdom 9000	. . . . . . . 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 3199	. . . . 5 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎)
47	10, 46	eqbrtrid 5183	. . . 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 3557	1 ⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068   ∪ cun 3961   ⊆ wss 3963  ∅c0 4339  {csn 4631  {cpr 4633  ⟨cop 4637   class class class wbr 5148  ^◡ccnv 5688  suc csuc 6388  Fun wfun 6557  ⟶wf 6559  –1-1→wf1 6560  ωcom 7887  1_oc1o 8498  2_oc2o 8499   ≼ cdom 8982   ≺ csdm 8983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-1o 8505  df-2o 8506  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988

This theorem is referenced by: (None)