Theorem indcardi 9955

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 8928	. . 3 ⊢ (𝑇 ∈ dom card → 𝑇 ≼ 𝑇)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝑇 ≼ 𝑇)
4		indcardi.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		cardon 9860	. . . 4 ⊢ (card‘𝑇) ∈ On
6	5	a1i 11	. . 3 ⊢ (𝜑 → (card‘𝑇) ∈ On)
7		simpl1 1193	. . . . 5 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒))) ∧ 𝑅 ≼ 𝑇) → 𝜑)
8		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒))) ∧ 𝑅 ≼ 𝑇) → 𝑅 ≼ 𝑇)
9		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑆 ≺ 𝑅)
10		simpl1 1193	. . . . . . . . . . . . . . . 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 8921	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ≺ 𝑅 → 𝑆 ≼ 𝑅)
13		simpl3 1195	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑅 ≼ 𝑇)
14		domtr 8948	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ≼ 𝑅 ∧ 𝑅 ≼ 𝑇) → 𝑆 ≼ 𝑇)
15	12, 13, 14	syl2an2 687	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑆 ≼ 𝑇)
16		numdom 9952	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom card ∧ 𝑆 ≼ 𝑇) → 𝑆 ∈ dom card)
17	11, 15, 16	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑆 ∈ dom card)
18		numdom 9952	. . . . . . . . . . . . . . 15 ⊢ ((𝑇 ∈ dom card ∧ 𝑅 ≼ 𝑇) → 𝑅 ∈ dom card)
19	11, 13, 18	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → 𝑅 ∈ dom card)
20		cardsdom2 9904	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ dom card ∧ 𝑅 ∈ dom card) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆 ≺ 𝑅))
21	17, 19, 20	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅 ≼ 𝑇) ∧ 𝑆 ≺ 𝑅) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆 ≺ 𝑅))
22	9, 21	mpbird 257	. . . . . . . . . . . 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 412	. . . . . . . . . 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 1918	. . . . . . . 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 1120	. . . . . . 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 1349	. . . . 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 1374	. . . 4 ⊢ (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆 ≼ 𝑇 → 𝜒))) ∧ 𝑅 ≼ 𝑇) → 𝜓)
34	33	ex 412	. . 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 5109	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑅 ≼ 𝑇 ↔ 𝑆 ≼ 𝑇))
37		indcardi.d	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
38	36, 37	imbi12d 344	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑅 ≼ 𝑇 → 𝜓) ↔ (𝑆 ≼ 𝑇 → 𝜒)))
39		indcardi.g	. . . . 5 ⊢ (𝑥 = 𝐴 → 𝑅 = 𝑇)
40	39	breq1d 5109	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑅 ≼ 𝑇 ↔ 𝑇 ≼ 𝑇))
41		indcardi.e	. . . 4 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜃))
42	40, 41	imbi12d 344	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑅 ≼ 𝑇 → 𝜓) ↔ (𝑇 ≼ 𝑇 → 𝜃)))
43	35	fveq2d 6839	. . 3 ⊢ (𝑥 = 𝑦 → (card‘𝑅) = (card‘𝑆))
44	39	fveq2d 6839	. . 3 ⊢ (𝑥 = 𝐴 → (card‘𝑅) = (card‘𝑇))
45	4, 6, 34, 38, 42, 43, 44	tfisi 7803	. 2 ⊢ (𝜑 → (𝑇 ≼ 𝑇 → 𝜃))
46	3, 45	mpd 15	1 ⊢ (𝜑 → 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114   ⊆ wss 3902   class class class wbr 5099  dom cdm 5625  Oncon0 6318  ‘cfv 6493   ≼ cdom 8885   ≺ csdm 8886  cardccrd 9851

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 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7317  df-ov 7363  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-er 8637  df-en 8888  df-dom 8889  df-sdom 8890  df-card 9855

This theorem is referenced by:  uzindi  13909  symggen  19403