Theorem inf3lem5 9561

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9564 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 7833	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ ω)
2	1	ancoms 458	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ω)
3		nnord 7830	. . . . . . 7 ⊢ (𝐴 ∈ ω → Ord 𝐴)
4		ordsucss 7773	. . . . . . 7 ⊢ (Ord 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
5	3, 4	syl 17	. . . . . 6 ⊢ (𝐴 ∈ ω → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
6	5	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 → suc 𝐵 ⊆ 𝐴))
7		peano2b 7839	. . . . . 6 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑣 = suc 𝐵 → (𝐹‘𝑣) = (𝐹‘suc 𝐵))
9	8	psseq2d 4055	. . . . . . . . 9 ⊢ (𝑣 = suc 𝐵 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))
10	9	imbi2d 340	. . . . . . . 8 ⊢ (𝑣 = suc 𝐵 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵))))
11		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
12	11	psseq2d 4055	. . . . . . . . 9 ⊢ (𝑣 = 𝑢 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘𝑢)))
13	12	imbi2d 340	. . . . . . . 8 ⊢ (𝑣 = 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑢))))
14		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
15	14	psseq2d 4055	. . . . . . . . 9 ⊢ (𝑣 = suc 𝑢 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)))
16	15	imbi2d 340	. . . . . . . 8 ⊢ (𝑣 = suc 𝑢 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑣)) ↔ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))))
17		fveq2 6840	. . . . . . . . . 10 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
18	17	psseq2d 4055	. . . . . . . . 9 ⊢ (𝑣 = 𝐴 → ((𝐹‘𝐵) ⊊ (𝐹‘𝑣) ↔ (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))
19	18	imbi2d 340	. . . . . . . 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 9560	. . . . . . . . . 10 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐵 ∈ ω → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))
24	23	com12 32	. . . . . . . . 9 ⊢ (𝐵 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))
25	7, 24	sylbir 235	. . . . . . . 8 ⊢ (suc 𝐵 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝐵)))
26		vex 3448	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
27	20, 21, 26, 22	inf3lem4 9560	. . . . . . . . . . 11 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝑢 ∈ ω → (𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢)))
28		psstr 4066	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝐵) ⊊ (𝐹‘𝑢) ∧ (𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢)) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))
29	28	expcom 413	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘suc 𝑢) → ((𝐹‘𝐵) ⊊ (𝐹‘𝑢) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢)))
30	27, 29	syl6com 37	. . . . . . . . . 10 ⊢ (𝑢 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐹‘𝐵) ⊊ (𝐹‘𝑢) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))))
31	30	a2d 29	. . . . . . . . 9 ⊢ (𝑢 ∈ ω → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝑢)) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘suc 𝑢))))
32	31	ad2antrr 726	. . . . . . . 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 7853	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ suc 𝐵 ∈ ω) ∧ suc 𝐵 ⊆ 𝐴) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))
34	33	ex 412	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ suc 𝐵 ∈ ω) → (suc 𝐵 ⊆ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
35	7, 34	sylan2b 594	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (suc 𝐵 ⊆ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
36	6, 35	syld 47	. . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ∈ 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
37	36	impancom 451	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐵 ∈ ω → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴))))
38	2, 37	mpd 15	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))
39	38	com12 32	1 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → (𝐹‘𝐵) ⊊ (𝐹‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  {crab 3402  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911   ⊊ wpss 3912  ∅c0 4292  ∪ cuni 4867   ↦ cmpt 5183   ↾ cres 5633  Ord word 6319  suc csuc 6322  ‘cfv 6499  ωcom 7822  reccrdg 8354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691  ax-reg 9521

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355

This theorem is referenced by:  inf3lem6  9562