Theorem findcard2d 7147

Description: Deduction version of findcard2 7145. If you also need 𝑦 ∈ Fin (which doesn't come for free due to ssfiexmid 7130), use findcard2sd 7148 instead. (Contributed by SO, 16-Jul-2018.)

Hypotheses

Ref	Expression
findcard2d.ch	⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))
findcard2d.th	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
findcard2d.ta	⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))
findcard2d.et	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
findcard2d.z	⊢ (𝜑 → 𝜒)
findcard2d.i	⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))
findcard2d.a	⊢ (𝜑 → 𝐴 ∈ Fin)

Assertion

Ref	Expression
findcard2d	⊢ (𝜑 → 𝜂)

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝜓,𝑦,𝑧   𝜒,𝑥   𝜃,𝑥   𝜏,𝑥   𝜂,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦,𝑧)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧)   𝜂(𝑦,𝑧)

Proof of Theorem findcard2d

Step	Hyp	Ref	Expression
1		ssid 3257	. 2 ⊢ 𝐴 ⊆ 𝐴
2		findcard2d.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin)
3	2	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝐴 ∈ Fin)
4		sseq1 3260	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
5	4	anbi2d 464	. . . . 5 ⊢ (𝑥 = ∅ → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
6		findcard2d.ch	. . . . 5 ⊢ (𝑥 = ∅ → (𝜓 ↔ 𝜒))
7	5, 6	imbi12d 234	. . . 4 ⊢ (𝑥 = ∅ → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)))
8		sseq1 3260	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
9	8	anbi2d 464	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑦 ⊆ 𝐴)))
10		findcard2d.th	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
11	9, 10	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃)))
12		sseq1 3260	. . . . . 6 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ 𝐴 ↔ (𝑦 ∪ {𝑧}) ⊆ 𝐴))
13	12	anbi2d 464	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)))
14		findcard2d.ta	. . . . 5 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝜓 ↔ 𝜏))
15	13, 14	imbi12d 234	. . . 4 ⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
16		sseq1 3260	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
17	16	anbi2d 464	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
18		findcard2d.et	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
19	17, 18	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ⊆ 𝐴) → 𝜓) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂)))
20		findcard2d.z	. . . . 5 ⊢ (𝜑 → 𝜒)
21	20	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐴) → 𝜒)
22		simprl 531	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝜑)
23		simprr 533	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
24	23	unssad 3395	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑦 ⊆ 𝐴)
25	22, 24	jca 306	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜑 ∧ 𝑦 ⊆ 𝐴))
26		id 19	. . . . . . . . . . 11 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → (𝑦 ∪ {𝑧}) ⊆ 𝐴)
27		vsnid 3720	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ {𝑧}
28		elun2 3386	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ {𝑧} → 𝑧 ∈ (𝑦 ∪ {𝑧}))
29	27, 28	mp1i 10	. . . . . . . . . . 11 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑧 ∈ (𝑦 ∪ {𝑧}))
30	26, 29	sseldd 3238	. . . . . . . . . 10 ⊢ ((𝑦 ∪ {𝑧}) ⊆ 𝐴 → 𝑧 ∈ 𝐴)
31	30	ad2antll 491	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴)
32		simplr 529	. . . . . . . . 9 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧 ∈ 𝑦)
33	31, 32	eldifd 3220	. . . . . . . 8 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ (𝐴 ∖ 𝑦))
34		findcard2d.i	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑦))) → (𝜃 → 𝜏))
35	22, 24, 33, 34	syl12anc 1272	. . . . . . 7 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (𝜃 → 𝜏))
36	25, 35	embantd 56	. . . . . 6 ⊢ (((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) ∧ (𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴)) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → 𝜏))
37	36	ex 115	. . . . 5 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → 𝜏)))
38	37	com23 78	. . . 4 ⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (((𝜑 ∧ 𝑦 ⊆ 𝐴) → 𝜃) → ((𝜑 ∧ (𝑦 ∪ {𝑧}) ⊆ 𝐴) → 𝜏)))
39	7, 11, 15, 19, 21, 38	findcard2s 7146	. . 3 ⊢ (𝐴 ∈ Fin → ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂))
40	3, 39	mpcom 36	. 2 ⊢ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → 𝜂)
41	1, 40	mpan2 425	1 ⊢ (𝜑 → 𝜂)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203   ∖ cdif 3207   ∪ cun 3208   ⊆ wss 3210  ∅c0 3507  {csn 3688  Fincfn 6974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-er 6766  df-en 6975  df-fin 6977

This theorem is referenced by:  iunfidisj  7212