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Mirrors > Home > MPE Home > Th. List > inf3lemd | Structured version Visualization version GIF version |
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8415 for detailed description. (Contributed by NM, 28-Oct-1996.) |
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 |
Ref | Expression |
---|---|
inf3lemd | ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6103 | . . . . 5 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = (𝐹‘∅)) | |
2 | inf3lem.1 | . . . . . 6 ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | |
3 | inf3lem.2 | . . . . . 6 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) | |
4 | inf3lem.3 | . . . . . 6 ⊢ 𝐴 ∈ V | |
5 | inf3lem.4 | . . . . . 6 ⊢ 𝐵 ∈ V | |
6 | 2, 3, 4, 5 | inf3lemb 8405 | . . . . 5 ⊢ (𝐹‘∅) = ∅ |
7 | 1, 6 | syl6eq 2660 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = ∅) |
8 | 0ss 3924 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
9 | 7, 8 | syl6eqss 3618 | . . 3 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) ⊆ 𝑥) |
10 | 9 | a1d 25 | . 2 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
11 | nnsuc 6974 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣) | |
12 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑣 ∈ V | |
13 | 2, 3, 12, 5 | inf3lemc 8406 | . . . . . . . . 9 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) = (𝐺‘(𝐹‘𝑣))) |
14 | 13 | eleq2d 2673 | . . . . . . . 8 ⊢ (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) ↔ 𝑢 ∈ (𝐺‘(𝐹‘𝑣)))) |
15 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑢 ∈ V | |
16 | fvex 6113 | . . . . . . . . . 10 ⊢ (𝐹‘𝑣) ∈ V | |
17 | 2, 3, 15, 16 | inf3lema 8404 | . . . . . . . . 9 ⊢ (𝑢 ∈ (𝐺‘(𝐹‘𝑣)) ↔ (𝑢 ∈ 𝑥 ∧ (𝑢 ∩ 𝑥) ⊆ (𝐹‘𝑣))) |
18 | 17 | simplbi 475 | . . . . . . . 8 ⊢ (𝑢 ∈ (𝐺‘(𝐹‘𝑣)) → 𝑢 ∈ 𝑥) |
19 | 14, 18 | syl6bi 242 | . . . . . . 7 ⊢ (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) → 𝑢 ∈ 𝑥)) |
20 | 19 | ssrdv 3574 | . . . . . 6 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥) |
21 | fveq2 6103 | . . . . . . 7 ⊢ (𝐴 = suc 𝑣 → (𝐹‘𝐴) = (𝐹‘suc 𝑣)) | |
22 | 21 | sseq1d 3595 | . . . . . 6 ⊢ (𝐴 = suc 𝑣 → ((𝐹‘𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥)) |
23 | 20, 22 | syl5ibrcom 236 | . . . . 5 ⊢ (𝑣 ∈ ω → (𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥)) |
24 | 23 | rexlimiv 3009 | . . . 4 ⊢ (∃𝑣 ∈ ω 𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥) |
25 | 11, 24 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ⊆ 𝑥) |
26 | 25 | expcom 450 | . 2 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
27 | 10, 26 | pm2.61ine 2865 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ↦ cmpt 4643 ↾ cres 5040 suc csuc 5642 ‘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 |
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: inf3lem2 8409 inf3lem3 8410 inf3lem6 8413 |
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