Theorem cardprclem 9965

Description: Lemma for cardprc 9966. (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 2861	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥})
3		abid 2751	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∣ (card‘𝑥) = 𝑥} ↔ (card‘𝑥) = 𝑥)
4		iscard 9961	. . . . . . . 8 ⊢ ((card‘𝑥) = 𝑥 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
5	2, 3, 4	3bitri 300	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 𝑦 ≺ 𝑥))
6	5	simplbi 501	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ On)
7	6	ssriv 3947	. . . . 5 ⊢ 𝐴 ⊆ On
8		ssonuni 7779	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 ⊆ On → ∪ 𝐴 ∈ On))
9	7, 8	mpi 21	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ On)
10		domrefg 8984	. . . . 5 ⊢ (∪ 𝐴 ∈ On → ∪ 𝐴 ≼ ∪ 𝐴)
11	9, 10	syl 18	. . . 4 ⊢ (𝐴 ∈ V → ∪ 𝐴 ≼ ∪ 𝐴)
12		elharval 9523	. . . 4 ⊢ (∪ 𝐴 ∈ (har‘∪ 𝐴) ↔ (∪ 𝐴 ∈ On ∧ ∪ 𝐴 ≼ ∪ 𝐴))
13	9, 11, 12	sylanbrc 594	. . 3 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ (har‘∪ 𝐴))
14	7	sseli 3939	. . . . . . . 8 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ On)
15		domrefg 8984	. . . . . . . . . 10 ⊢ (𝑧 ∈ On → 𝑧 ≼ 𝑧)
16	15	ancli 557	. . . . . . . . 9 ⊢ (𝑧 ∈ On → (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
17		elharval 9523	. . . . . . . . 9 ⊢ (𝑧 ∈ (har‘𝑧) ↔ (𝑧 ∈ On ∧ 𝑧 ≼ 𝑧))
18	16, 17	sylibr 237	. . . . . . . 8 ⊢ (𝑧 ∈ On → 𝑧 ∈ (har‘𝑧))
19	14, 18	syl 18	. . . . . . 7 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ (har‘𝑧))
20		harcard 9964	. . . . . . . 8 ⊢ (card‘(har‘𝑧)) = (har‘𝑧)
21		fvex 6895	. . . . . . . . 9 ⊢ (har‘𝑧) ∈ V
22		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → (card‘𝑥) = (card‘(har‘𝑧)))
23		id 23	. . . . . . . . . 10 ⊢ (𝑥 = (har‘𝑧) → 𝑥 = (har‘𝑧))
24	22, 23	eqeq12d 2785	. . . . . . . . 9 ⊢ (𝑥 = (har‘𝑧) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘𝑧)) = (har‘𝑧)))
25	21, 24, 1	elab2 3648	. . . . . . . 8 ⊢ ((har‘𝑧) ∈ 𝐴 ↔ (card‘(har‘𝑧)) = (har‘𝑧))
26	20, 25	mpbir 234	. . . . . . 7 ⊢ (har‘𝑧) ∈ 𝐴
27		eleq2 2858	. . . . . . . . 9 ⊢ (𝑤 = (har‘𝑧) → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ (har‘𝑧)))
28		eleq1 2857	. . . . . . . . 9 ⊢ (𝑤 = (har‘𝑧) → (𝑤 ∈ 𝐴 ↔ (har‘𝑧) ∈ 𝐴))
29	27, 28	anbi12d 643	. . . . . . . 8 ⊢ (𝑤 = (har‘𝑧) → ((𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴) ↔ (𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴)))
30	21, 29	spcev 3572	. . . . . . 7 ⊢ ((𝑧 ∈ (har‘𝑧) ∧ (har‘𝑧) ∈ 𝐴) → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
31	19, 26, 30	sylancl 597	. . . . . 6 ⊢ (𝑧 ∈ 𝐴 → ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
32		eluni 4877	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝐴 ↔ ∃𝑤(𝑧 ∈ 𝑤 ∧ 𝑤 ∈ 𝐴))
33	31, 32	sylibr 237	. . . . 5 ⊢ (𝑧 ∈ 𝐴 → 𝑧 ∈ ∪ 𝐴)
34	33	ssriv 3947	. . . 4 ⊢ 𝐴 ⊆ ∪ 𝐴
35		harcard 9964	. . . . 5 ⊢ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)
36		fvex 6895	. . . . . 6 ⊢ (har‘∪ 𝐴) ∈ V
37		fveq2 6882	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → (card‘𝑥) = (card‘(har‘∪ 𝐴)))
38		id 23	. . . . . . 7 ⊢ (𝑥 = (har‘∪ 𝐴) → 𝑥 = (har‘∪ 𝐴))
39	37, 38	eqeq12d 2785	. . . . . 6 ⊢ (𝑥 = (har‘∪ 𝐴) → ((card‘𝑥) = 𝑥 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴)))
40	36, 39, 1	elab2 3648	. . . . 5 ⊢ ((har‘∪ 𝐴) ∈ 𝐴 ↔ (card‘(har‘∪ 𝐴)) = (har‘∪ 𝐴))
41	35, 40	mpbir 234	. . . 4 ⊢ (har‘∪ 𝐴) ∈ 𝐴
42	34, 41	sselii 3940	. . 3 ⊢ (har‘∪ 𝐴) ∈ ∪ 𝐴
43	13, 42	jctir 529	. 2 ⊢ (𝐴 ∈ V → (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
44		eloni 6371	. . 3 ⊢ (∪ 𝐴 ∈ On → Ord ∪ 𝐴)
45		ordn2lp 6381	. . 3 ⊢ (Ord ∪ 𝐴 → ¬ (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
46	9, 44, 45	3syl 19	. 2 ⊢ (𝐴 ∈ V → ¬ (∪ 𝐴 ∈ (har‘∪ 𝐴) ∧ (har‘∪ 𝐴) ∈ ∪ 𝐴))
47	43, 46	pm2.65i 196	1 ⊢ ¬ 𝐴 ∈ V

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  Vcvv 3461   ⊆ wss 3911  ∪ cuni 4874   class class class wbr 5111  Ord word 6360  Oncon0 6361  ‘cfv 6537   ≼ cdom 8941   ≺ csdm 8942  harchar 9518  cardccrd 9921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-oi 9472  df-har 9519  df-card 9925

This theorem is referenced by:  cardprc  9966