Theorem indcardi 9797

Description: Indirect strong induction on the cardinality of a finite or numerable set. (Contributed by Stefan O'Rear, 24-Aug-2015.)

Hypotheses

Ref	Expression
indcardi.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
indcardi.b	⊢ (𝜑 → 𝑇 ∈ dom card)
indcardi.c	⊢ ((𝜑 ∧ 𝑅 ≼ 𝑇 ∧ ∀𝑦(𝑆 ≺ 𝑅 → 𝜒)) → 𝜓)
indcardi.d	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
indcardi.e	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
indcardi.f	⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)
indcardi.g	⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)

Assertion

Ref	Expression
indcardi	⊢ (𝜑 → 𝜃)

Distinct variable groups:   𝑥,𝑦,𝑇   𝑥,𝐴   𝑥,𝑆   𝜒,𝑥   𝜑,𝑥,𝑦   𝜃,𝑥   𝑦,𝑅   𝜓,𝑦

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)   𝐴(𝑦)   𝑅(𝑥)   𝑆(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem indcardi

Step	Hyp	Ref	Expression
1		indcardi.b	. . 3 ⊢ (𝜑 → 𝑇 ∈ dom card)
2		domrefg 8775	. . 3 ⊢ (𝑇 ∈ dom card → 𝑇 ≼ 𝑇)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑇 ≼ 𝑇)
4		indcardi.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		cardon 9702	. . . 4 ⊢ (card‘𝑇) ∈ On
6	5	a1i 11	. . 3 ⊢ (𝜑 → (card‘𝑇) ∈ On)
7		simpl1 1190	. . . . 5 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒))) ∧ 𝑅 ≼ 𝑇) → 𝜑)
8		simpr 485	. . . . 5 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒))) ∧ 𝑅 ≼ 𝑇) → 𝑅 ≼ 𝑇)
9		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑆 ≺ 𝑅)
10		simpl1 1190	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝜑)
11	10, 1	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑇 ∈ dom card)
12		sdomdom 8768	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ≺ 𝑅 → 𝑆 ≼ 𝑅)
13		simpl3 1192	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑅 ≼ 𝑇)
14		domtr 8793	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ≼ 𝑅 ∧ 𝑅 ≼ 𝑇) → 𝑆 ≼ 𝑇)
15	12, 13, 14	syl2an2 683	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑆 ≼ 𝑇)
16		numdom 9794	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom card ∧ 𝑆 ≼ 𝑇) → 𝑆 ∈ dom card)
17	11, 15, 16	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑆 ∈ dom card)
18		numdom 9794	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom card ∧ 𝑅 ≼ 𝑇) → 𝑅 ∈ dom card)
19	11, 13, 18	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑅 ∈ dom card)
20		cardsdom2 9746	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ dom card ∧ 𝑅 ∈ dom card) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆 ≺ 𝑅))
21	17, 19, 20	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆 ≺ 𝑅))
22	9, 21	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → (card‘𝑆) ∈ (card‘𝑅))
23		id 22	. . . . . . . . . . . . 13 ⊢ (((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)) → ((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)))
24	23	com3l 89	. . . . . . . . . . . 12 ⊢ ((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)) → 𝜒)))
25	22, 15, 24	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)) → 𝜒))
26	25	ex 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) → (𝑆 ≺ 𝑅 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)) → 𝜒)))
27	26	com23 86	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)) → (𝑆 ≺ 𝑅 → 𝜒)))
28	27	alimdv 1919	. . . . . . . 8 ⊢ ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)) → ∀𝑦(𝑆 ≺ 𝑅 → 𝜒)))
29	28	3exp 1118	. . . . . . 7 ⊢ (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (𝑅 ≼ 𝑇 → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)) → ∀𝑦(𝑆 ≺ 𝑅 → 𝜒)))))
30	29	com34 91	. . . . . 6 ⊢ (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒)) → (𝑅 ≼ 𝑇 → ∀𝑦(𝑆 ≺ 𝑅 → 𝜒)))))
31	30	3imp1 1346	. . . . 5 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒))) ∧ 𝑅 ≼ 𝑇) → ∀𝑦(𝑆 ≺ 𝑅 → 𝜒))
32		indcardi.c	. . . . 5 ⊢ ((𝜑 ∧ 𝑅 ≼ 𝑇 ∧ ∀𝑦(𝑆 ≺ 𝑅 → 𝜒)) → 𝜓)
33	7, 8, 31, 32	syl3anc 1370	. . . 4 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒))) ∧ 𝑅 ≼ 𝑇) → 𝜓)
34	33	ex 413	. . 3 ⊢ ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒))) → (𝑅 ≼ 𝑇 → 𝜓))
35		indcardi.f	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝑅 = 𝑆)
36	35	breq1d 5084	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑅 ≼ 𝑇 ↔ 𝑆 ≼ 𝑇))
37		indcardi.d	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
38	36, 37	imbi12d 345	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑅 ≼ 𝑇 → 𝜓) ↔ (𝑆 ≼ 𝑇 → 𝜒)))
39		indcardi.g	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)
40	39	breq1d 5084	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑅 ≼ 𝑇 ↔ 𝑇 ≼ 𝑇))
41		indcardi.e	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
42	40, 41	imbi12d 345	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑅 ≼ 𝑇 → 𝜓) ↔ (𝑇 ≼ 𝑇 → 𝜃)))
43	35	fveq2d 6778	. . 3 ⊢ (𝑥 = 𝑦 → (card‘𝑅) = (card‘𝑆))
44	39	fveq2d 6778	. . 3 ⊢ (𝑥 = 𝐴 → (card‘𝑅) = (card‘𝑇))
45	4, 6, 34, 38, 42, 43, 44	tfisi 7705	. 2 ⊢ (𝜑 → (𝑇 ≼ 𝑇 → 𝜃))
46	3, 45	mpd 15	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887   class class class wbr 5074  dom cdm 5589  Oncon0 6266  ‘cfv 6433   ≼ cdom 8731   ≺ csdm 8732  cardccrd 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-card 9697

This theorem is referenced by:  uzindi  13702  symggen  19078