Theorem 1sdom 8857

Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8685.) (Contributed by Mario Carneiro, 12-Jan-2013.)

Assertion

Ref	Expression
1sdom	⊢ (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 1sdom

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

Step	Hyp	Ref	Expression
1		breq2 5043	. 2 ⊢ (𝑎 = 𝐴 → (1o ≺ 𝑎 ↔ 1o ≺ 𝐴))
2		rexeq 3310	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
3	2	rexeqbi1dv 3308	. 2 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
4		1onn 8345	. . . 4 ⊢ 1o ∈ ω
5		sucdom 8851	. . . 4 ⊢ (1o ∈ ω → (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎))
6	4, 5	ax-mp 5	. . 3 ⊢ (1o ≺ 𝑎 ↔ suc 1o ≼ 𝑎)
7		df-2o 8181	. . . 4 ⊢ 2o = suc 1o
8	7	breq1i 5046	. . 3 ⊢ (2o ≼ 𝑎 ↔ suc 1o ≼ 𝑎)
9		2dom 8685	. . . 4 ⊢ (2o ≼ 𝑎 → ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
10		df2o3 8195	. . . . 5 ⊢ 2o = {∅, 1o}
11		vex 3402	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		vex 3402	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
13		0ex 5185	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14		1oex 8193	. . . . . . . . . . . 12 ⊢ 1o ∈ V
15	11, 12, 13, 14	funpr 6414	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
16		df-ne 2933	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
17		1n0 8199	. . . . . . . . . . . . . . 15 ⊢ 1o ≠ ∅
18	17	necomi 2986	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1o
19	13, 14, 11, 12	fpr 6947	. . . . . . . . . . . . . 14 ⊢ (∅ ≠ 1o → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦})
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}⟶{𝑥, 𝑦}
21		df-f1 6363	. . . . . . . . . . . . 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 6069	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
24	14, 12	cnvsn 6069	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨1o, 𝑦⟩} = {⟨𝑦, 1o⟩}
25	23, 24	uneq12i 4061	. . . . . . . . . . . . . 14 ⊢ (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
26		df-pr 4530	. . . . . . . . . . . . . . . 16 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
27	26	cnveqi 5728	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩})
28		cnvun 5986	. . . . . . . . . . . . . . 15 ⊢ ◡({⟨∅, 𝑥⟩} ∪ {⟨1o, 𝑦⟩}) = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
29	27, 28	eqtri 2759	. . . . . . . . . . . . . 14 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1o, 𝑦⟩})
30		df-pr 4530	. . . . . . . . . . . . . 14 ⊢ {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1o⟩})
31	25, 29, 30	3eqtr4i 2769	. . . . . . . . . . . . 13 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩}
32	31	funeqi 6379	. . . . . . . . . . . 12 ⊢ (Fun ◡{⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ↔ Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩})
33	22, 32	bitr2i 279	. . . . . . . . . . 11 ⊢ (Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1o⟩} ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
34	15, 16, 33	3imtr3i 294	. . . . . . . . . 10 ⊢ (¬ 𝑥 = 𝑦 → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦})
35		prssi 4720	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36		f1ss 6599	. . . . . . . . . 10 ⊢ (({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎)
37	34, 35, 36	syl2an 599	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎)
38		prex 5310	. . . . . . . . . 10 ⊢ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ V
39		f1eq1 6588	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} → (𝑓:{∅, 1o}–1-1→𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}:{∅, 1o}–1-1→𝑎))
40	38, 39	spcev 3511	. . . . . . . . 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 3402	. . . . . . . . 9 ⊢ 𝑎 ∈ V
43	42	brdom 8618	. . . . . . . 8 ⊢ ({∅, 1o} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1o}–1-1→𝑎)
44	41, 43	sylibr 237	. . . . . . 7 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {∅, 1o} ≼ 𝑎)
45	44	expcom 417	. . . . . 6 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎))
46	45	rexlimivv 3201	. . . . 5 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → {∅, 1o} ≼ 𝑎)
47	10, 46	eqbrtrid 5074	. . . 4 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → 2o ≼ 𝑎)
48	9, 47	impbii 212	. . 3 ⊢ (2o ≼ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
49	6, 8, 48	3bitr2i 302	. 2 ⊢ (1o ≺ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
50	1, 3, 49	vtoclbg 3473	1 ⊢ (𝐴 ∈ 𝑉 → (1o ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∃wex 1787   ∈ wcel 2112   ≠ wne 2932  ∃wrex 3052   ∪ cun 3851   ⊆ wss 3853  ∅c0 4223  {csn 4527  {cpr 4529  ⟨cop 4533   class class class wbr 5039  ^◡ccnv 5535  suc csuc 6193  Fun wfun 6352  ⟶wf 6354  –1-1→wf1 6355  ωcom 7622  1_oc1o 8173  2_oc2o 8174   ≼ cdom 8602   ≺ csdm 8603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-om 7623  df-1o 8180  df-2o 8181  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607

This theorem is referenced by:  unxpdomlem3  8860  frgpnabl  19214  isnzr2  20255