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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 9092 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 6664 | . . . . 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 9082 | . . . . 5 ⊢ (𝐹‘∅) = ∅ |
7 | 1, 6 | syl6eq 2872 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = ∅) |
8 | 0ss 4349 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
9 | 7, 8 | eqsstrdi 4020 | . . 3 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) ⊆ 𝑥) |
10 | 9 | a1d 25 | . 2 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
11 | nnsuc 7591 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣) | |
12 | vex 3497 | . . . . . . . . . 10 ⊢ 𝑣 ∈ V | |
13 | 2, 3, 12, 5 | inf3lemc 9083 | . . . . . . . . 9 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) = (𝐺‘(𝐹‘𝑣))) |
14 | 13 | eleq2d 2898 | . . . . . . . 8 ⊢ (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) ↔ 𝑢 ∈ (𝐺‘(𝐹‘𝑣)))) |
15 | vex 3497 | . . . . . . . . . 10 ⊢ 𝑢 ∈ V | |
16 | fvex 6677 | . . . . . . . . . 10 ⊢ (𝐹‘𝑣) ∈ V | |
17 | 2, 3, 15, 16 | inf3lema 9081 | . . . . . . . . 9 ⊢ (𝑢 ∈ (𝐺‘(𝐹‘𝑣)) ↔ (𝑢 ∈ 𝑥 ∧ (𝑢 ∩ 𝑥) ⊆ (𝐹‘𝑣))) |
18 | 17 | simplbi 500 | . . . . . . . 8 ⊢ (𝑢 ∈ (𝐺‘(𝐹‘𝑣)) → 𝑢 ∈ 𝑥) |
19 | 14, 18 | syl6bi 255 | . . . . . . 7 ⊢ (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) → 𝑢 ∈ 𝑥)) |
20 | 19 | ssrdv 3972 | . . . . . 6 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥) |
21 | fveq2 6664 | . . . . . . 7 ⊢ (𝐴 = suc 𝑣 → (𝐹‘𝐴) = (𝐹‘suc 𝑣)) | |
22 | 21 | sseq1d 3997 | . . . . . 6 ⊢ (𝐴 = suc 𝑣 → ((𝐹‘𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥)) |
23 | 20, 22 | syl5ibrcom 249 | . . . . 5 ⊢ (𝑣 ∈ ω → (𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥)) |
24 | 23 | rexlimiv 3280 | . . . 4 ⊢ (∃𝑣 ∈ ω 𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥) |
25 | 11, 24 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ⊆ 𝑥) |
26 | 25 | expcom 416 | . 2 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
27 | 10, 26 | pm2.61ine 3100 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 ↦ cmpt 5138 ↾ cres 5551 suc csuc 6187 ‘cfv 6349 ωcom 7574 reccrdg 8039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 |
This theorem is referenced by: inf3lem2 9086 inf3lem3 9087 inf3lem6 9090 |
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