Theorem 1sdom 8163

Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 8029.) (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 4657	. 2 ⊢ (𝑎 = 𝐴 → (1𝑜 ≺ 𝑎 ↔ 1𝑜 ≺ 𝐴))
2		rexeq 3139	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
3	2	rexeqbi1dv 3147	. 2 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
4		1onn 7719	. . . 4 ⊢ 1𝑜 ∈ ω
5		sucdom 8157	. . . 4 ⊢ (1𝑜 ∈ ω → (1𝑜 ≺ 𝑎 ↔ suc 1𝑜 ≼ 𝑎))
6	4, 5	ax-mp 5	. . 3 ⊢ (1𝑜 ≺ 𝑎 ↔ suc 1𝑜 ≼ 𝑎)
7		df-2o 7561	. . . 4 ⊢ 2𝑜 = suc 1𝑜
8	7	breq1i 4660	. . 3 ⊢ (2𝑜 ≼ 𝑎 ↔ suc 1𝑜 ≼ 𝑎)
9		2dom 8029	. . . 4 ⊢ (2𝑜 ≼ 𝑎 → ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
10		df2o3 7573	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
11		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		vex 3203	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
13		0ex 4790	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14	4	elexi 3213	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ V
15	11, 12, 13, 14	funpr 5944	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩})
16		df-ne 2795	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
17		1n0 7575	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 ≠ ∅
18	17	necomi 2848	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1𝑜
19	13, 14, 11, 12	fpr 6421	. . . . . . . . . . . . . 14 ⊢ (∅ ≠ 1𝑜 → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦})
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦}
21		df-f1 5893	. . . . . . . . . . . . 13 ⊢ ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ↔ ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦} ∧ Fun ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}))
22	20, 21	mpbiran 953	. . . . . . . . . . . 12 ⊢ ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ↔ Fun ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩})
23	13, 11	cnvsn 5618	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
24	14, 12	cnvsn 5618	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨1𝑜, 𝑦⟩} = {⟨𝑦, 1𝑜⟩}
25	23, 24	uneq12i 3765	. . . . . . . . . . . . . 14 ⊢ (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1𝑜, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1𝑜⟩})
26		df-pr 4180	. . . . . . . . . . . . . . . 16 ⊢ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
27	26	cnveqi 5297	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ◡({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
28		cnvun 5538	. . . . . . . . . . . . . . 15 ⊢ ◡({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}) = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1𝑜, 𝑦⟩})
29	27, 28	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1𝑜, 𝑦⟩})
30		df-pr 4180	. . . . . . . . . . . . . 14 ⊢ {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1𝑜⟩})
31	25, 29, 30	3eqtr4i 2654	. . . . . . . . . . . . 13 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩}
32	31	funeqi 5909	. . . . . . . . . . . 12 ⊢ (Fun ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} ↔ Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩})
33	22, 32	bitr2i 265	. . . . . . . . . . 11 ⊢ (Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩} ↔ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦})
34	15, 16, 33	3imtr3i 280	. . . . . . . . . 10 ⊢ (¬ 𝑥 = 𝑦 → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦})
35		prssi 4353	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36		f1ss 6106	. . . . . . . . . 10 ⊢ (({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→𝑎)
37	34, 35, 36	syl2an 494	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→𝑎)
38		prex 4909	. . . . . . . . . 10 ⊢ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} ∈ V
39		f1eq1 6096	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} → (𝑓:{∅, 1𝑜}–1-1→𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→𝑎))
40	38, 39	spcev 3300	. . . . . . . . 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 3203	. . . . . . . . 9 ⊢ 𝑎 ∈ V
43	42	brdom 7967	. . . . . . . 8 ⊢ ({∅, 1𝑜} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1𝑜}–1-1→𝑎)
44	41, 43	sylibr 224	. . . . . . 7 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {∅, 1𝑜} ≼ 𝑎)
45	44	expcom 451	. . . . . 6 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1𝑜} ≼ 𝑎))
46	45	rexlimivv 3036	. . . . 5 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → {∅, 1𝑜} ≼ 𝑎)
47	10, 46	syl5eqbr 4688	. . . 4 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → 2𝑜 ≼ 𝑎)
48	9, 47	impbii 199	. . 3 ⊢ (2𝑜 ≼ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
49	6, 8, 48	3bitr2i 288	. 2 ⊢ (1𝑜 ≺ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
50	1, 3, 49	vtoclbg 3267	1 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  {cpr 4179  ⟨cop 4183   class class class wbr 4653  ^◡ccnv 5113  suc csuc 5725  Fun wfun 5882  ⟶wf 5884  –1-1→wf1 5885  ωcom 7065  1_𝑜c1o 7553  2_𝑜c2o 7554   ≼ cdom 7953   ≺ csdm 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958

This theorem is referenced by:  unxpdomlem3  8166  frgpnabl  18278  isnzr2  19263