Theorem inf3lem6 8413

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 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
inf3lem6	⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)

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 inf3lem6

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

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . . . . . . . 11 ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦})
2		inf3lem.2	. . . . . . . . . . 11 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω)
3		vex 3176	. . . . . . . . . . 11 ⊢ 𝑢 ∈ V
4		vex 3176	. . . . . . . . . . 11 ⊢ 𝑣 ∈ V
5	1, 2, 3, 4	inf3lem5 8412	. . . . . . . . . 10 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑢 ∈ ω ∧ 𝑣 ∈ 𝑢) → (𝐹‘𝑣) ⊊ (𝐹‘𝑢)))
6		dfpss2 3654	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑣) ⊊ (𝐹‘𝑢) ↔ ((𝐹‘𝑣) ⊆ (𝐹‘𝑢) ∧ ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
7	6	simprbi 479	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣) ⊊ (𝐹‘𝑢) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢))
8	5, 7	syl6 34	. . . . . . . . 9 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑢 ∈ ω ∧ 𝑣 ∈ 𝑢) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
9	8	expdimp 452	. . . . . . . 8 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ 𝑢 ∈ ω) → (𝑣 ∈ 𝑢 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
10	9	adantrl 748	. . . . . . 7 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 ∈ 𝑢 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
11	1, 2, 4, 3	inf3lem5 8412	. . . . . . . . . 10 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑣 ∈ ω ∧ 𝑢 ∈ 𝑣) → (𝐹‘𝑢) ⊊ (𝐹‘𝑣)))
12		dfpss2 3654	. . . . . . . . . . . 12 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘𝑣) ↔ ((𝐹‘𝑢) ⊆ (𝐹‘𝑣) ∧ ¬ (𝐹‘𝑢) = (𝐹‘𝑣)))
13	12	simprbi 479	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘𝑣) → ¬ (𝐹‘𝑢) = (𝐹‘𝑣))
14		eqcom 2617	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑢) = (𝐹‘𝑣) ↔ (𝐹‘𝑣) = (𝐹‘𝑢))
15	13, 14	sylnib 317	. . . . . . . . . 10 ⊢ ((𝐹‘𝑢) ⊊ (𝐹‘𝑣) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢))
16	11, 15	syl6 34	. . . . . . . . 9 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ((𝑣 ∈ ω ∧ 𝑢 ∈ 𝑣) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
17	16	expdimp 452	. . . . . . . 8 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ 𝑣 ∈ ω) → (𝑢 ∈ 𝑣 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
18	17	adantrr 749	. . . . . . 7 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑢 ∈ 𝑣 → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
19	10, 18	jaod 394	. . . . . 6 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣) → ¬ (𝐹‘𝑣) = (𝐹‘𝑢)))
20	19	con2d 128	. . . . 5 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹‘𝑣) = (𝐹‘𝑢) → ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
21		nnord 6965	. . . . . . 7 ⊢ (𝑣 ∈ ω → Ord 𝑣)
22		nnord 6965	. . . . . . 7 ⊢ (𝑢 ∈ ω → Ord 𝑢)
23		ordtri3 5676	. . . . . . 7 ⊢ ((Ord 𝑣 ∧ Ord 𝑢) → (𝑣 = 𝑢 ↔ ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
24	21, 22, 23	syl2an 493	. . . . . 6 ⊢ ((𝑣 ∈ ω ∧ 𝑢 ∈ ω) → (𝑣 = 𝑢 ↔ ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
25	24	adantl 481	. . . . 5 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → (𝑣 = 𝑢 ↔ ¬ (𝑣 ∈ 𝑢 ∨ 𝑢 ∈ 𝑣)))
26	20, 25	sylibrd 248	. . . 4 ⊢ (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ∧ (𝑣 ∈ ω ∧ 𝑢 ∈ ω)) → ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢))
27	26	ralrimivva 2954	. . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢))
28		frfnom 7417	. . . . . 6 ⊢ (rec(𝐺, ∅) ↾ ω) Fn ω
29		fneq1 5893	. . . . . 6 ⊢ (𝐹 = (rec(𝐺, ∅) ↾ ω) → (𝐹 Fn ω ↔ (rec(𝐺, ∅) ↾ ω) Fn ω))
30	28, 29	mpbiri 247	. . . . 5 ⊢ (𝐹 = (rec(𝐺, ∅) ↾ ω) → 𝐹 Fn ω)
31		fvelrnb 6153	. . . . . . . 8 ⊢ (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 ↔ ∃𝑣 ∈ ω (𝐹‘𝑣) = 𝑢))
32		inf3lem.4	. . . . . . . . . . . 12 ⊢ 𝐵 ∈ V
33	1, 2, 4, 32	inf3lemd 8407	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ω → (𝐹‘𝑣) ⊆ 𝑥)
34		fvex 6113	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑣) ∈ V
35	34	elpw 4114	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑣) ∈ 𝒫 𝑥 ↔ (𝐹‘𝑣) ⊆ 𝑥)
36	33, 35	sylibr 223	. . . . . . . . . 10 ⊢ (𝑣 ∈ ω → (𝐹‘𝑣) ∈ 𝒫 𝑥)
37		eleq1 2676	. . . . . . . . . 10 ⊢ ((𝐹‘𝑣) = 𝑢 → ((𝐹‘𝑣) ∈ 𝒫 𝑥 ↔ 𝑢 ∈ 𝒫 𝑥))
38	36, 37	syl5ibcom 234	. . . . . . . . 9 ⊢ (𝑣 ∈ ω → ((𝐹‘𝑣) = 𝑢 → 𝑢 ∈ 𝒫 𝑥))
39	38	rexlimiv 3009	. . . . . . . 8 ⊢ (∃𝑣 ∈ ω (𝐹‘𝑣) = 𝑢 → 𝑢 ∈ 𝒫 𝑥)
40	31, 39	syl6bi 242	. . . . . . 7 ⊢ (𝐹 Fn ω → (𝑢 ∈ ran 𝐹 → 𝑢 ∈ 𝒫 𝑥))
41	40	ssrdv 3574	. . . . . 6 ⊢ (𝐹 Fn ω → ran 𝐹 ⊆ 𝒫 𝑥)
42	41	ancli 572	. . . . 5 ⊢ (𝐹 Fn ω → (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
43	2, 30, 42	mp2b 10	. . . 4 ⊢ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥)
44		df-f 5808	. . . 4 ⊢ (𝐹:ω⟶𝒫 𝑥 ↔ (𝐹 Fn ω ∧ ran 𝐹 ⊆ 𝒫 𝑥))
45	43, 44	mpbir 220	. . 3 ⊢ 𝐹:ω⟶𝒫 𝑥
46	27, 45	jctil 558	. 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢)))
47		dff13 6416	. 2 ⊢ (𝐹:ω–1-1→𝒫 𝑥 ↔ (𝐹:ω⟶𝒫 𝑥 ∧ ∀𝑣 ∈ ω ∀𝑢 ∈ ω ((𝐹‘𝑣) = (𝐹‘𝑢) → 𝑣 = 𝑢)))
48	46, 47	sylibr 223	1 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → 𝐹:ω–1-1→𝒫 𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   ⊊ wpss 3541  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040  Ord word 5639   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  ωcom 6957  reccrdg 7392

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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380

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-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-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-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-lim 5645  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-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393

This theorem is referenced by:  inf3lem7  8414  dominf  9150  dominfac  9274