Theorem inf3lem5 9541

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9544 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 7819	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝐵 ∈ ω)
2	1	ancoms 458	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ ω)
3		nnord 7816	. . . . . . 7 ⊢ (𝐴 ∈ ω → Ord 𝐴)
4		ordsucss 7760	. . . . . . 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 7825	. . . . . 6 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω)
8		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑣 = suc 𝐵 → (𝐹‘𝑣) = (𝐹‘suc 𝐵))
9	8	psseq2d 4048	. . . . . . . . 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 6834	. . . . . . . . . 10 ⊢ (𝑣 = 𝑢 → (𝐹‘𝑣) = (𝐹‘𝑢))
12	11	psseq2d 4048	. . . . . . . . 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 6834	. . . . . . . . . 10 ⊢ (𝑣 = suc 𝑢 → (𝐹‘𝑣) = (𝐹‘suc 𝑢))
15	14	psseq2d 4048	. . . . . . . . 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 6834	. . . . . . . . . 10 ⊢ (𝑣 = 𝐴 → (𝐹‘𝑣) = (𝐹‘𝐴))
18	17	psseq2d 4048	. . . . . . . . 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 9540	. . . . . . . . . 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 3444	. . . . . . . . . . . 12 ⊢ 𝑢 ∈ V
27	20, 21, 26, 22	inf3lem4 9540	. . . . . . . . . . 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 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 7839	. . . . . . 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 1541   ∈ wcel 2113   ≠ wne 2932  {crab 3399  Vcvv 3440   ∩ cin 3900   ⊆ wss 3901   ⊊ wpss 3902  ∅c0 4285  ∪ cuni 4863   ↦ cmpt 5179   ↾ cres 5626  Ord word 6316  suc csuc 6319  ‘cfv 6492  ωcom 7808  reccrdg 8340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680  ax-reg 9497

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341

This theorem is referenced by:  inf3lem6  9542