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Theorem indcardi 9261
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 (𝜑𝐴𝑉)
indcardi.b (𝜑𝑇 ∈ dom card)
indcardi.c ((𝜑𝑅𝑇 ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
indcardi.d (𝑥 = 𝑦 → (𝜓𝜒))
indcardi.e (𝑥 = 𝐴 → (𝜓𝜃))
indcardi.f (𝑥 = 𝑦𝑅 = 𝑆)
indcardi.g (𝑥 = 𝐴𝑅 = 𝑇)
Assertion
Ref Expression
indcardi (𝜑𝜃)
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
StepHypRef Expression
1 indcardi.b . . 3 (𝜑𝑇 ∈ dom card)
2 domrefg 8341 . . 3 (𝑇 ∈ dom card → 𝑇𝑇)
31, 2syl 17 . 2 (𝜑𝑇𝑇)
4 indcardi.a . . 3 (𝜑𝐴𝑉)
5 cardon 9167 . . . 4 (card‘𝑇) ∈ On
65a1i 11 . . 3 (𝜑 → (card‘𝑇) ∈ On)
7 simpl1 1171 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝜑)
8 simpr 477 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝑅𝑇)
9 simpr 477 . . . . . . . . . . . . 13 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑅)
10 simpl1 1171 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝜑)
1110, 1syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑇 ∈ dom card)
12 sdomdom 8334 . . . . . . . . . . . . . . . 16 (𝑆𝑅𝑆𝑅)
13 simpl3 1173 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑅𝑇)
14 domtr 8359 . . . . . . . . . . . . . . . 16 ((𝑆𝑅𝑅𝑇) → 𝑆𝑇)
1512, 13, 14syl2an2 673 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑇)
16 numdom 9258 . . . . . . . . . . . . . . 15 ((𝑇 ∈ dom card ∧ 𝑆𝑇) → 𝑆 ∈ dom card)
1711, 15, 16syl2anc 576 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆 ∈ dom card)
18 numdom 9258 . . . . . . . . . . . . . . 15 ((𝑇 ∈ dom card ∧ 𝑅𝑇) → 𝑅 ∈ dom card)
1911, 13, 18syl2anc 576 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑅 ∈ dom card)
20 cardsdom2 9211 . . . . . . . . . . . . . 14 ((𝑆 ∈ dom card ∧ 𝑅 ∈ dom card) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆𝑅))
2117, 19, 20syl2anc 576 . . . . . . . . . . . . 13 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆𝑅))
229, 21mpbird 249 . . . . . . . . . . . 12 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → (card‘𝑆) ∈ (card‘𝑅))
23 id 22 . . . . . . . . . . . . 13 (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)))
2423com3l 89 . . . . . . . . . . . 12 ((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒)))
2522, 15, 24sylc 65 . . . . . . . . . . 11 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒))
2625ex 405 . . . . . . . . . 10 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (𝑆𝑅 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒)))
2726com23 86 . . . . . . . . 9 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → (𝑆𝑅𝜒)))
2827alimdv 1875 . . . . . . . 8 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ∀𝑦(𝑆𝑅𝜒)))
29283exp 1099 . . . . . . 7 (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (𝑅𝑇 → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ∀𝑦(𝑆𝑅𝜒)))))
3029com34 91 . . . . . 6 (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → (𝑅𝑇 → ∀𝑦(𝑆𝑅𝜒)))))
31303imp1 1327 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → ∀𝑦(𝑆𝑅𝜒))
32 indcardi.c . . . . 5 ((𝜑𝑅𝑇 ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
337, 8, 31, 32syl3anc 1351 . . . 4 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝜓)
3433ex 405 . . 3 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) → (𝑅𝑇𝜓))
35 indcardi.f . . . . 5 (𝑥 = 𝑦𝑅 = 𝑆)
3635breq1d 4939 . . . 4 (𝑥 = 𝑦 → (𝑅𝑇𝑆𝑇))
37 indcardi.d . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
3836, 37imbi12d 337 . . 3 (𝑥 = 𝑦 → ((𝑅𝑇𝜓) ↔ (𝑆𝑇𝜒)))
39 indcardi.g . . . . 5 (𝑥 = 𝐴𝑅 = 𝑇)
4039breq1d 4939 . . . 4 (𝑥 = 𝐴 → (𝑅𝑇𝑇𝑇))
41 indcardi.e . . . 4 (𝑥 = 𝐴 → (𝜓𝜃))
4240, 41imbi12d 337 . . 3 (𝑥 = 𝐴 → ((𝑅𝑇𝜓) ↔ (𝑇𝑇𝜃)))
4335fveq2d 6503 . . 3 (𝑥 = 𝑦 → (card‘𝑅) = (card‘𝑆))
4439fveq2d 6503 . . 3 (𝑥 = 𝐴 → (card‘𝑅) = (card‘𝑇))
454, 6, 34, 38, 42, 43, 44tfisi 7389 . 2 (𝜑 → (𝑇𝑇𝜃))
463, 45mpd 15 1 (𝜑𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068  wal 1505   = wceq 1507  wcel 2050  wss 3829   class class class wbr 4929  dom cdm 5407  Oncon0 6029  cfv 6188  cdom 8304  csdm 8305  cardccrd 9158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-wrecs 7750  df-recs 7812  df-er 8089  df-en 8307  df-dom 8308  df-sdom 8309  df-card 9162
This theorem is referenced by:  uzindi  13165  symggen  18359
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