Theorem 1sdom 8048

Description: A set that strictly dominates ordinal 1 has at least 2 different members. (Closely related to 2dom 7915.) (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 4587	. 2 ⊢ (𝑎 = 𝐴 → (1𝑜 ≺ 𝑎 ↔ 1𝑜 ≺ 𝐴))
2		rexeq 3116	. . 3 ⊢ (𝑎 = 𝐴 → (∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
3	2	rexeqbi1dv 3124	. 2 ⊢ (𝑎 = 𝐴 → (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))
4		1onn 7606	. . . 4 ⊢ 1𝑜 ∈ ω
5		sucdom 8042	. . . 4 ⊢ (1𝑜 ∈ ω → (1𝑜 ≺ 𝑎 ↔ suc 1𝑜 ≼ 𝑎))
6	4, 5	ax-mp 5	. . 3 ⊢ (1𝑜 ≺ 𝑎 ↔ suc 1𝑜 ≼ 𝑎)
7		df-2o 7448	. . . 4 ⊢ 2𝑜 = suc 1𝑜
8	7	breq1i 4590	. . 3 ⊢ (2𝑜 ≼ 𝑎 ↔ suc 1𝑜 ≼ 𝑎)
9		2dom 7915	. . . 4 ⊢ (2𝑜 ≼ 𝑎 → ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
10		df2o3 7460	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
11		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
12		vex 3176	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
13		0ex 4718	. . . . . . . . . . . 12 ⊢ ∅ ∈ V
14	4	elexi 3186	. . . . . . . . . . . 12 ⊢ 1𝑜 ∈ V
15	11, 12, 13, 14	funpr 5858	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 → Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩})
16		df-ne 2782	. . . . . . . . . . 11 ⊢ (𝑥 ≠ 𝑦 ↔ ¬ 𝑥 = 𝑦)
17		1n0 7462	. . . . . . . . . . . . . . 15 ⊢ 1𝑜 ≠ ∅
18	17	necomi 2836	. . . . . . . . . . . . . 14 ⊢ ∅ ≠ 1𝑜
19	13, 14, 11, 12	fpr 6326	. . . . . . . . . . . . . 14 ⊢ (∅ ≠ 1𝑜 → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦})
20	18, 19	ax-mp 5	. . . . . . . . . . . . 13 ⊢ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦}
21		df-f1 5809	. . . . . . . . . . . . 13 ⊢ ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ↔ ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}⟶{𝑥, 𝑦} ∧ Fun ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}))
22	20, 21	mpbiran 955	. . . . . . . . . . . 12 ⊢ ({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ↔ Fun ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩})
23	13, 11	cnvsn 5536	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩} = {⟨𝑥, ∅⟩}
24	14, 12	cnvsn 5536	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨1𝑜, 𝑦⟩} = {⟨𝑦, 1𝑜⟩}
25	23, 24	uneq12i 3727	. . . . . . . . . . . . . 14 ⊢ (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1𝑜, 𝑦⟩}) = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1𝑜⟩})
26		df-pr 4128	. . . . . . . . . . . . . . . 16 ⊢ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
27	26	cnveqi 5219	. . . . . . . . . . . . . . 15 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = ◡({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩})
28		cnvun 5457	. . . . . . . . . . . . . . 15 ⊢ ◡({⟨∅, 𝑥⟩} ∪ {⟨1𝑜, 𝑦⟩}) = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1𝑜, 𝑦⟩})
29	27, 28	eqtri 2632	. . . . . . . . . . . . . 14 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = (◡{⟨∅, 𝑥⟩} ∪ ◡{⟨1𝑜, 𝑦⟩})
30		df-pr 4128	. . . . . . . . . . . . . 14 ⊢ {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩} = ({⟨𝑥, ∅⟩} ∪ {⟨𝑦, 1𝑜⟩})
31	25, 29, 30	3eqtr4i 2642	. . . . . . . . . . . . 13 ⊢ ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} = {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩}
32	31	funeqi 5824	. . . . . . . . . . . 12 ⊢ (Fun ◡{⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} ↔ Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩})
33	22, 32	bitr2i 264	. . . . . . . . . . 11 ⊢ (Fun {⟨𝑥, ∅⟩, ⟨𝑦, 1𝑜⟩} ↔ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦})
34	15, 16, 33	3imtr3i 279	. . . . . . . . . 10 ⊢ (¬ 𝑥 = 𝑦 → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦})
35		prssi 4293	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → {𝑥, 𝑦} ⊆ 𝑎)
36		f1ss 6019	. . . . . . . . . 10 ⊢ (({⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑎) → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→𝑎)
37	34, 35, 36	syl2an 493	. . . . . . . . 9 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→𝑎)
38		prex 4836	. . . . . . . . . 10 ⊢ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} ∈ V
39		f1eq1 6009	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩} → (𝑓:{∅, 1𝑜}–1-1→𝑎 ↔ {⟨∅, 𝑥⟩, ⟨1𝑜, 𝑦⟩}:{∅, 1𝑜}–1-1→𝑎))
40	38, 39	spcev 3273	. . . . . . . . 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 3176	. . . . . . . . 9 ⊢ 𝑎 ∈ V
43	42	brdom 7853	. . . . . . . 8 ⊢ ({∅, 1𝑜} ≼ 𝑎 ↔ ∃𝑓 𝑓:{∅, 1𝑜}–1-1→𝑎)
44	41, 43	sylibr 223	. . . . . . 7 ⊢ ((¬ 𝑥 = 𝑦 ∧ (𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎)) → {∅, 1𝑜} ≼ 𝑎)
45	44	expcom 450	. . . . . 6 ⊢ ((𝑥 ∈ 𝑎 ∧ 𝑦 ∈ 𝑎) → (¬ 𝑥 = 𝑦 → {∅, 1𝑜} ≼ 𝑎))
46	45	rexlimivv 3018	. . . . 5 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → {∅, 1𝑜} ≼ 𝑎)
47	10, 46	syl5eqbr 4618	. . . 4 ⊢ (∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦 → 2𝑜 ≼ 𝑎)
48	9, 47	impbii 198	. . 3 ⊢ (2𝑜 ≼ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
49	6, 8, 48	3bitr2i 287	. 2 ⊢ (1𝑜 ≺ 𝑎 ↔ ∃𝑥 ∈ 𝑎 ∃𝑦 ∈ 𝑎 ¬ 𝑥 = 𝑦)
50	1, 3, 49	vtoclbg 3240	1 ⊢ (𝐴 ∈ 𝑉 → (1𝑜 ≺ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 ¬ 𝑥 = 𝑦))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  ^◡ccnv 5037  suc csuc 5642  Fun wfun 5798  ⟶wf 5800  –1-1→wf1 5801  ωcom 6957  1_𝑜c1o 7440  2_𝑜c2o 7441   ≼ cdom 7839   ≺ csdm 7840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-2o 7448  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844

This theorem is referenced by:  unxpdomlem3  8051  frgpnabl  18101  isnzr2  19084