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Theorem indcardi 8747
 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 7876 . . 3 (𝑇 ∈ dom card → 𝑇𝑇)
31, 2syl 17 . 2 (𝜑𝑇𝑇)
4 indcardi.a . . 3 (𝜑𝐴𝑉)
5 cardon 8653 . . . 4 (card‘𝑇) ∈ On
65a1i 11 . . 3 (𝜑 → (card‘𝑇) ∈ On)
7 simpl1 1057 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝜑)
8 simpr 476 . . . . 5 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝑅𝑇)
9 simpr 476 . . . . . . . . . . . . 13 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑅)
10 simpl1 1057 . . . . . . . . . . . . . . . 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 7869 . . . . . . . . . . . . . . . . 17 (𝑆𝑅𝑆𝑅)
1312adantl 481 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑅)
14 simpl3 1059 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑅𝑇)
15 domtr 7895 . . . . . . . . . . . . . . . 16 ((𝑆𝑅𝑅𝑇) → 𝑆𝑇)
1613, 14, 15syl2anc 691 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆𝑇)
17 numdom 8744 . . . . . . . . . . . . . . 15 ((𝑇 ∈ dom card ∧ 𝑆𝑇) → 𝑆 ∈ dom card)
1811, 16, 17syl2anc 691 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑆 ∈ dom card)
19 numdom 8744 . . . . . . . . . . . . . . 15 ((𝑇 ∈ dom card ∧ 𝑅𝑇) → 𝑅 ∈ dom card)
2011, 14, 19syl2anc 691 . . . . . . . . . . . . . 14 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → 𝑅 ∈ dom card)
21 cardsdom2 8697 . . . . . . . . . . . . . 14 ((𝑆 ∈ dom card ∧ 𝑅 ∈ dom card) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆𝑅))
2218, 20, 21syl2anc 691 . . . . . . . . . . . . 13 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → ((card‘𝑆) ∈ (card‘𝑅) ↔ 𝑆𝑅))
239, 22mpbird 246 . . . . . . . . . . . 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 87 . . . . . . . . . . . 12 ((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒)))
2623, 16, 25sylc 63 . . . . . . . . . . 11 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) ∧ 𝑆𝑅) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒))
2726ex 449 . . . . . . . . . 10 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (𝑆𝑅 → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → 𝜒)))
2827com23 84 . . . . . . . . 9 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → (𝑆𝑅𝜒)))
2928alimdv 1832 . . . . . . . 8 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ 𝑅𝑇) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ∀𝑦(𝑆𝑅𝜒)))
30293exp 1256 . . . . . . 7 (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (𝑅𝑇 → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → ∀𝑦(𝑆𝑅𝜒)))))
3130com34 89 . . . . . 6 (𝜑 → (((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) → (∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒)) → (𝑅𝑇 → ∀𝑦(𝑆𝑅𝜒)))))
32313imp1 1272 . . . . 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 1318 . . . 4 (((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) ∧ 𝑅𝑇) → 𝜓)
3534ex 449 . . 3 ((𝜑 ∧ ((card‘𝑅) ∈ On ∧ (card‘𝑅) ⊆ (card‘𝑇)) ∧ ∀𝑦((card‘𝑆) ∈ (card‘𝑅) → (𝑆𝑇𝜒))) → (𝑅𝑇𝜓))
36 indcardi.f . . . . 5 (𝑥 = 𝑦𝑅 = 𝑆)
3736breq1d 4593 . . . 4 (𝑥 = 𝑦 → (𝑅𝑇𝑆𝑇))
38 indcardi.d . . . 4 (𝑥 = 𝑦 → (𝜓𝜒))
3937, 38imbi12d 333 . . 3 (𝑥 = 𝑦 → ((𝑅𝑇𝜓) ↔ (𝑆𝑇𝜒)))
40 indcardi.g . . . . 5 (𝑥 = 𝐴𝑅 = 𝑇)
4140breq1d 4593 . . . 4 (𝑥 = 𝐴 → (𝑅𝑇𝑇𝑇))
42 indcardi.e . . . 4 (𝑥 = 𝐴 → (𝜓𝜃))
4341, 42imbi12d 333 . . 3 (𝑥 = 𝐴 → ((𝑅𝑇𝜓) ↔ (𝑇𝑇𝜃)))
4436fveq2d 6107 . . 3 (𝑥 = 𝑦 → (card‘𝑅) = (card‘𝑆))
4540fveq2d 6107 . . 3 (𝑥 = 𝐴 → (card‘𝑅) = (card‘𝑇))
464, 6, 35, 39, 43, 44, 45tfisi 6950 . 2 (𝜑 → (𝑇𝑇𝜃))
473, 46mpd 15 1 (𝜑𝜃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   class class class wbr 4583  dom cdm 5038  Oncon0 5640  ‘cfv 5804   ≼ cdom 7839   ≺ csdm 7840  cardccrd 8644 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648 This theorem is referenced by:  uzindi  12643  symggen  17713
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