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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 9627 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 6889 | . . . . 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 9617 | . . . . 5 ⊢ (𝐹‘∅) = ∅ |
7 | 1, 6 | eqtrdi 2789 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = ∅) |
8 | 0ss 4396 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
9 | 7, 8 | eqsstrdi 4036 | . . 3 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) ⊆ 𝑥) |
10 | 9 | a1d 25 | . 2 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
11 | nnsuc 7870 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣) | |
12 | vex 3479 | . . . . . . . . . 10 ⊢ 𝑣 ∈ V | |
13 | 2, 3, 12, 5 | inf3lemc 9618 | . . . . . . . . 9 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) = (𝐺‘(𝐹‘𝑣))) |
14 | 13 | eleq2d 2820 | . . . . . . . 8 ⊢ (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) ↔ 𝑢 ∈ (𝐺‘(𝐹‘𝑣)))) |
15 | vex 3479 | . . . . . . . . . 10 ⊢ 𝑢 ∈ V | |
16 | fvex 6902 | . . . . . . . . . 10 ⊢ (𝐹‘𝑣) ∈ V | |
17 | 2, 3, 15, 16 | inf3lema 9616 | . . . . . . . . 9 ⊢ (𝑢 ∈ (𝐺‘(𝐹‘𝑣)) ↔ (𝑢 ∈ 𝑥 ∧ (𝑢 ∩ 𝑥) ⊆ (𝐹‘𝑣))) |
18 | 17 | simplbi 499 | . . . . . . . 8 ⊢ (𝑢 ∈ (𝐺‘(𝐹‘𝑣)) → 𝑢 ∈ 𝑥) |
19 | 14, 18 | syl6bi 253 | . . . . . . 7 ⊢ (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) → 𝑢 ∈ 𝑥)) |
20 | 19 | ssrdv 3988 | . . . . . 6 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥) |
21 | fveq2 6889 | . . . . . . 7 ⊢ (𝐴 = suc 𝑣 → (𝐹‘𝐴) = (𝐹‘suc 𝑣)) | |
22 | 21 | sseq1d 4013 | . . . . . 6 ⊢ (𝐴 = suc 𝑣 → ((𝐹‘𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥)) |
23 | 20, 22 | syl5ibrcom 246 | . . . . 5 ⊢ (𝑣 ∈ ω → (𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥)) |
24 | 23 | rexlimiv 3149 | . . . 4 ⊢ (∃𝑣 ∈ ω 𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥) |
25 | 11, 24 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ⊆ 𝑥) |
26 | 25 | expcom 415 | . 2 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
27 | 10, 26 | pm2.61ine 3026 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 {crab 3433 Vcvv 3475 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ↦ cmpt 5231 ↾ cres 5678 suc csuc 6364 ‘cfv 6541 ωcom 7852 reccrdg 8406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 |
This theorem is referenced by: inf3lem2 9621 inf3lem3 9622 inf3lem6 9625 |
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