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Theorem indcardi 8809
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 7935 . . 3 (𝑇 ∈ dom card → 𝑇𝑇)
31, 2syl 17 . 2 (𝜑𝑇𝑇)
4 indcardi.a . . 3 (𝜑𝐴𝑉)
5 cardon 8715 . . . 4 (card‘𝑇) ∈ On
65a1i 11 . . 3 (𝜑 → (card‘𝑇) ∈ On)
7 simpl1 1062 . . . . 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 1062 . . . . . . . . . . . . . . . 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 7928 . . . . . . . . . . . . . . . . 17 (𝑆𝑅𝑆𝑅)
1312adantl 482 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑅)
14 simpl3 1064 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑅𝑇)
15 domtr 7954 . . . . . . . . . . . . . . . 16 ((𝑆𝑅𝑅𝑇) → 𝑆𝑇)
1613, 14, 15syl2anc 692 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑇)
17 numdom 8806 . . . . . . . . . . . . . . 15 ((𝑇 ∈ dom card ∧ 𝑆𝑇) → 𝑆 ∈ dom card)
1811, 16, 17syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆 ∈ dom card)
19 numdom 8806 . . . . . . . . . . . . . . 15 ((𝑇 ∈ dom card ∧ 𝑅𝑇) → 𝑅 ∈ dom card)
2011, 14, 19syl2anc 692 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑅 ∈ dom card)
21 cardsdom2 8759 . . . . . . . . . . . . . 14 ((𝑆 ∈ dom card ∧ 𝑅 ∈ dom card) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆𝑅))
2218, 20, 21syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆𝑅))
239, 22mpbird 247 . . . . . . . . . . . 12 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → (card‘𝑆) ∈ (card‘𝑅))
24 id 22 . . . . . . . . . . . . 13 (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)))
2524com3l 89 . . . . . . . . . . . 12 ((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒)))
2623, 16, 25sylc 65 . . . . . . . . . . 11 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒))
2726ex 450 . . . . . . . . . 10 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (𝑆𝑅 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒)))
2827com23 86 . . . . . . . . 9 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → (𝑆𝑅𝜒)))
2928alimdv 1847 . . . . . . . 8 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ∀𝑦(𝑆𝑅𝜒)))
30293exp 1261 . . . . . . 7 (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (𝑅𝑇 → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ∀𝑦(𝑆𝑅𝜒)))))
3130com34 91 . . . . . 6 (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → (𝑅𝑇 → ∀𝑦(𝑆𝑅𝜒)))))
32313imp1 1277 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → ∀𝑦(𝑆𝑅𝜒))
33 indcardi.c . . . . 5 ((𝜑𝑅𝑇 ∧ ∀𝑦(𝑆𝑅𝜒)) → 𝜓)
347, 8, 32, 33syl3anc 1323 . . . 4 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝜓)
3534ex 450 . . 3 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) → (𝑅𝑇𝜓))
36 indcardi.f . . . . 5 (𝑥 = 𝑦𝑅 = 𝑆)
3736breq1d 4628 . . . 4 (𝑥 = 𝑦 → (𝑅𝑇𝑆𝑇))
38 indcardi.d . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
3937, 38imbi12d 334 . . 3 (𝑥 = 𝑦 → ((𝑅𝑇𝜓) ↔ (𝑆𝑇𝜒)))
40 indcardi.g . . . . 5 (𝑥 = 𝐴𝑅 = 𝑇)
4140breq1d 4628 . . . 4 (𝑥 = 𝐴 → (𝑅𝑇𝑇𝑇))
42 indcardi.e . . . 4 (𝑥 = 𝐴 → (𝜓𝜃))
4341, 42imbi12d 334 . . 3 (𝑥 = 𝐴 → ((𝑅𝑇𝜓) ↔ (𝑇𝑇𝜃)))
4436fveq2d 6154 . . 3 (𝑥 = 𝑦 → (card‘𝑅) = (card‘𝑆))
4540fveq2d 6154 . . 3 (𝑥 = 𝐴 → (card‘𝑅) = (card‘𝑇))
464, 6, 35, 39, 43, 44, 45tfisi 7006 . 2 (𝜑 → (𝑇𝑇𝜃))
473, 46mpd 15 1 (𝜑𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1992  wss 3560   class class class wbr 4618  dom cdm 5079  Oncon0 5685  cfv 5850  cdom 7898  csdm 7899  cardccrd 8706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-wrecs 7353  df-recs 7414  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-card 8710
This theorem is referenced by:  uzindi  12718  symggen  17806
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