Theorem cardprclem 9903

Description: Lemma for cardprc 9904. (Contributed by Mario Carneiro, 22-Jan-2013.) (Revised by Mario Carneiro, 15-May-2015.)

Hypothesis

Ref	Expression
cardprclem.1	⊢ 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥}

Assertion

Ref	Expression
cardprclem	⊢ ¬ 𝐴 ∈ V

Distinct variable group:   𝑥,𝐴

Proof of Theorem cardprclem

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cardprclem.1	. . . . . . . . 9 ⊢ 𝐴 = {𝑥 ∣ (card‘𝑥) = 𝑥}
2	1	eleq2i 2828	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥})
3		abid 2718	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4		iscard 9899	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
5	2, 3, 4	3bitri 297	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
6	5	simplbi 496	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
7	6	ssriv 3925	. . . . 5 ⊢ 𝐴 ⊆ On
8		ssonuni 7734	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
9	7, 8	mpi 20	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ On)
10		domrefg 8934	. . . . 5 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 ≼ ∪ 𝐴)
11	9, 10	syl 17	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ≼ ∪ 𝐴)
12		elharval 9476	. . . 4 ⊢ (∪ 𝐴 ∈ (har‘∪ 𝐴) ↔ (∪ 𝐴 ∈ On ∧ ∪ 𝐴 ≼ ∪ 𝐴))
13	9, 11, 12	sylanbrc 584	. . 3 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ (har‘∪ 𝐴))
14	7	sseli 3917	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ On)
15		domrefg 8934	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → 𝑧 ≼ 𝑧)
16	15	ancli 548	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
17		elharval 9476	. . . . . . . . 9 ⊢ (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
18	16, 17	sylibr 234	. . . . . . . 8 ⊢ (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
19	14, 18	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (har‘𝑧))
20		harcard 9902	. . . . . . . 8 ⊢ (card‘(har‘𝑧)) = (har‘𝑧)
21		fvex 6853	. . . . . . . . 9 ⊢ (har‘𝑧) ∈ V
22		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23		id 22	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
24	22, 23	eqeq12d 2752	. . . . . . . . 9 ⊢ (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
25	21, 24, 1	elab2 3625	. . . . . . . 8 ⊢ ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
26	20, 25	mpbir 231	. . . . . . 7 ⊢ (har‘𝑧) ∈ 𝐴
27		eleq2 2825	. . . . . . . . 9 ⊢ (𝑤 = (har‘𝑧) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (har‘𝑧)))
28		eleq1 2824	. . . . . . . . 9 ⊢ (𝑤 = (har‘𝑧) → (𝑤 ∈ 𝐴 ↔ (har‘𝑧) ∈ 𝐴))
29	27, 28	anbi12d 633	. . . . . . . 8 ⊢ (𝑤 = (har‘𝑧) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
30	21, 29	spcev 3548	. . . . . . 7 ⊢ ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
31	19, 26, 30	sylancl 587	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
32		eluni 4853	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
33	31, 32	sylibr 234	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴)
34	33	ssriv 3925	. . . 4 ⊢ 𝐴 ⊆ ∪ 𝐴
35		harcard 9902	. . . . 5 ⊢ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)
36		fvex 6853	. . . . . 6 ⊢ (har‘∪ 𝐴) ∈ V
37		fveq2 6840	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → (card‘𝑥) = (card‘(har‘∪ 𝐴)))
38		id 22	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → 𝑥 = (har‘∪ 𝐴))
39	37, 38	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = (har‘∪ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)))
40	36, 39, 1	elab2 3625	. . . . 5 ⊢ ((har‘∪ 𝐴) ∈ 𝐴 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴))
41	35, 40	mpbir 231	. . . 4 ⊢ (har‘∪ 𝐴) ∈ 𝐴
42	34, 41	sselii 3918	. . 3 ⊢ (har‘∪ 𝐴) ∈ ∪ 𝐴
43	13, 42	jctir 520	. 2 ⊢ (𝐴 ∈ V → (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
44		eloni 6333	. . 3 ⊢ (∪ 𝐴 ∈ On → Ord ∪ 𝐴)
45		ordn2lp 6343	. . 3 ⊢ (Ord ∪ 𝐴 → ¬ (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
46	9, 44, 45	3syl 18	. 2 ⊢ (𝐴 ∈ V → ¬ (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
47	43, 46	pm2.65i 194	1 ⊢ ¬ 𝐴 ∈ V

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∀wral 3051  Vcvv 3429   ⊆ wss 3889  ∪ cuni 4850   class class class wbr 5085  Ord word 6322  Oncon0 6323  ‘cfv 6498   ≼ cdom 8891   ≺ csdm 8892  harchar 9471  cardccrd 9859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-oi 9425  df-har 9472  df-card 9863

This theorem is referenced by:  cardprc  9904