Theorem inf3lem5 9615

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9618 for detailed description. (Contributed by NM, 29-Oct-1996.)

Hypotheses

Ref	Expression
inf3lem.1	⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
inf3lem.2	⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)
inf3lem.3	⊢ 𝐴 ∈ V
inf3lem.4	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
inf3lem5	⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))

Distinct variable group:   𝑥,𝑦,𝑤

Allowed substitution hints:   𝐴(𝑥, 𝑦, 𝑤)   𝐵(𝑥, 𝑦, 𝑤)   𝐹(𝑥, 𝑦, 𝑤)   𝐺(𝑥, 𝑦, 𝑤)

Proof of Theorem inf3lem5

Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elnn 7877	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ ω)
2	1	ancoms 464	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ω)
3		nnord 7874	. . . . . . 7 ⊢ (𝐴 ∈ ω → Ord 𝐴)
4		ordsucss 7818	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
5	3, 4	syl 18	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
6	5	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
7		peano2b 7883	. . . . . 6 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑣 = suc 𝐵 → (𝐹‘𝑣) = (𝐹‘suc 𝐵))
9	8	psseq2d 4047	. . . . . . . . 9 ⊢ (𝑣 = suc 𝐵 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))
10	9	imbi2d 343	. . . . . . . 8 ⊢ (𝑣 = suc 𝐵 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵))))
11		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
12	11	psseq2d 4047	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘𝑢)))
13	12	imbi2d 343	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑢))))
14		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
15	14	psseq2d 4047	. . . . . . . . 9 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)))
16	15	imbi2d 343	. . . . . . . 8 ⊢ (𝑣 = suc 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))))
17		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
18	17	psseq2d 4047	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))
19	18	imbi2d 343	. . . . . . . 8 ⊢ (𝑣 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
20		inf3lem.1	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
21		inf3lem.2	. . . . . . . . . . 11 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)
22		inf3lem.4	. . . . . . . . . . 11 ⊢ 𝐵 ∈ V
23	20, 21, 22, 22	inf3lem4 9614	. . . . . . . . . 10 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐵 ∈ ω → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))
24	23	com12 33	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))
25	7, 24	sylbir 238	. . . . . . . 8 ⊢ (suc 𝐵 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))
26		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
27	20, 21, 26, 22	inf3lem4 9614	. . . . . . . . . . 11 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑢 ∈ ω → (𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢)))
28		psstr 4059	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝐵) ⊊ (𝐹‘𝑢) ∧ (𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢)) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))
29	28	expcom 419	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢) → ((𝐹‘𝐵) ⊊ (𝐹‘𝑢) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)))
30	27, 29	syl6com 38	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐹‘𝐵) ⊊ (𝐹‘𝑢) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))))
31	30	a2d 30	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑢)) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))))
32	31	ad2antrr 739	. . . . . . . 8 ⊢ (((𝑢 ∈ ω ∧ suc 𝐵 ∈ ω) ∧ suc 𝐵 ⊆ 𝑢) → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑢)) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))))
33	10, 13, 16, 19, 25, 32	findsg 7898	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ suc 𝐵 ∈ ω) ∧ suc 𝐵 ⊆ 𝐴) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))
34	33	ex 418	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ suc 𝐵 ∈ ω) → (suc 𝐵 ⊆ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
35	7, 34	sylan2b 606	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐵 ⊆ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
36	6, 35	syld 48	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
37	36	impancom 457	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
38	2, 37	mpd 16	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))
39	38	com12 33	1 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  {crab 3414  Vcvv 3453   ∩ cin 3901   ⊆ wss 3902   ⊊ wpss 3903  ∅c0 4282  ∪ cuni 4870   ↦ cmpt 5190   ↾ cres 5661  Ord word 6360  suc csuc 6363  ‘cfv 6537  ωcom 7866  reccrdg 8402

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740  ax-reg 9568

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403

This theorem is used by:  inf3lem6  9616