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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 9082 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 6645 | . . . . 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 9072 | . . . . 5 ⊢ (𝐹‘∅) = ∅ |
7 | 1, 6 | eqtrdi 2849 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = ∅) |
8 | 0ss 4304 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
9 | 7, 8 | eqsstrdi 3969 | . . 3 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) ⊆ 𝑥) |
10 | 9 | a1d 25 | . 2 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
11 | nnsuc 7577 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣) | |
12 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑣 ∈ V | |
13 | 2, 3, 12, 5 | inf3lemc 9073 | . . . . . . . . 9 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) = (𝐺‘(𝐹‘𝑣))) |
14 | 13 | eleq2d 2875 | . . . . . . . 8 ⊢ (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) ↔ 𝑢 ∈ (𝐺‘(𝐹‘𝑣)))) |
15 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑢 ∈ V | |
16 | fvex 6658 | . . . . . . . . . 10 ⊢ (𝐹‘𝑣) ∈ V | |
17 | 2, 3, 15, 16 | inf3lema 9071 | . . . . . . . . 9 ⊢ (𝑢 ∈ (𝐺‘(𝐹‘𝑣)) ↔ (𝑢 ∈ 𝑥 ∧ (𝑢 ∩ 𝑥) ⊆ (𝐹‘𝑣))) |
18 | 17 | simplbi 501 | . . . . . . . 8 ⊢ (𝑢 ∈ (𝐺‘(𝐹‘𝑣)) → 𝑢 ∈ 𝑥) |
19 | 14, 18 | syl6bi 256 | . . . . . . 7 ⊢ (𝑣 ∈ ω → (𝑢 ∈ (𝐹‘suc 𝑣) → 𝑢 ∈ 𝑥)) |
20 | 19 | ssrdv 3921 | . . . . . 6 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥) |
21 | fveq2 6645 | . . . . . . 7 ⊢ (𝐴 = suc 𝑣 → (𝐹‘𝐴) = (𝐹‘suc 𝑣)) | |
22 | 21 | sseq1d 3946 | . . . . . 6 ⊢ (𝐴 = suc 𝑣 → ((𝐹‘𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥)) |
23 | 20, 22 | syl5ibrcom 250 | . . . . 5 ⊢ (𝑣 ∈ ω → (𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥)) |
24 | 23 | rexlimiv 3239 | . . . 4 ⊢ (∃𝑣 ∈ ω 𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥) |
25 | 11, 24 | syl 17 | . . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → (𝐹‘𝐴) ⊆ 𝑥) |
26 | 25 | expcom 417 | . 2 ⊢ (𝐴 ≠ ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
27 | 10, 26 | pm2.61ine 3070 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∃wrex 3107 {crab 3110 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ↦ cmpt 5110 ↾ cres 5521 suc csuc 6161 ‘cfv 6324 ωcom 7560 reccrdg 8028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 |
This theorem is referenced by: inf3lem2 9076 inf3lem3 9077 inf3lem6 9080 |
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