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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 9579 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 6846 | . . . . 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 9569 | . . . . 5 ⊢ (𝐹‘∅) = ∅ |
7 | 1, 6 | eqtrdi 2789 | . . . 4 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) = ∅) |
8 | 0ss 4360 | . . . 4 ⊢ ∅ ⊆ 𝑥 | |
9 | 7, 8 | eqsstrdi 4002 | . . 3 ⊢ (𝐴 = ∅ → (𝐹‘𝐴) ⊆ 𝑥) |
10 | 9 | a1d 25 | . 2 ⊢ (𝐴 = ∅ → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥)) |
11 | nnsuc 7824 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ 𝐴 ≠ ∅) → ∃𝑣 ∈ ω 𝐴 = suc 𝑣) | |
12 | vex 3451 | . . . . . . . . . 10 ⊢ 𝑣 ∈ V | |
13 | 2, 3, 12, 5 | inf3lemc 9570 | . . . . . . . . 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 3451 | . . . . . . . . . 10 ⊢ 𝑢 ∈ V | |
16 | fvex 6859 | . . . . . . . . . 10 ⊢ (𝐹‘𝑣) ∈ V | |
17 | 2, 3, 15, 16 | inf3lema 9568 | . . . . . . . . 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 3954 | . . . . . 6 ⊢ (𝑣 ∈ ω → (𝐹‘suc 𝑣) ⊆ 𝑥) |
21 | fveq2 6846 | . . . . . . 7 ⊢ (𝐴 = suc 𝑣 → (𝐹‘𝐴) = (𝐹‘suc 𝑣)) | |
22 | 21 | sseq1d 3979 | . . . . . 6 ⊢ (𝐴 = suc 𝑣 → ((𝐹‘𝐴) ⊆ 𝑥 ↔ (𝐹‘suc 𝑣) ⊆ 𝑥)) |
23 | 20, 22 | syl5ibrcom 247 | . . . . 5 ⊢ (𝑣 ∈ ω → (𝐴 = suc 𝑣 → (𝐹‘𝐴) ⊆ 𝑥)) |
24 | 23 | rexlimiv 3142 | . . . 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 3025 | 1 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∃wrex 3070 {crab 3406 Vcvv 3447 ∩ cin 3913 ⊆ wss 3914 ∅c0 4286 ↦ cmpt 5192 ↾ cres 5639 suc csuc 6323 ‘cfv 6500 ωcom 7806 reccrdg 8359 |
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 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 |
This theorem is referenced by: inf3lem2 9573 inf3lem3 9574 inf3lem6 9577 |
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