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Mirrors > Home > MPE Home > Th. List > inf3lem4 | Structured version Visualization version GIF version |
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9393 for detailed description. (Contributed by NM, 29-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 |
---|---|
inf3lem4 | ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ⊊ (𝐹‘suc 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inf3lem.1 | . . . . 5 ⊢ 𝐺 = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | |
2 | inf3lem.2 | . . . . 5 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) | |
3 | inf3lem.3 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | inf3lem.4 | . . . . 5 ⊢ 𝐵 ∈ V | |
5 | 1, 2, 3, 4 | inf3lem1 9386 | . . . 4 ⊢ (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴)) |
6 | 5 | a1i 11 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴))) |
7 | 1, 2, 3, 4 | inf3lem3 9388 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ≠ (𝐹‘suc 𝐴))) |
8 | 6, 7 | jcad 513 | . 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → ((𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴) ∧ (𝐹‘𝐴) ≠ (𝐹‘suc 𝐴)))) |
9 | df-pss 3906 | . 2 ⊢ ((𝐹‘𝐴) ⊊ (𝐹‘suc 𝐴) ↔ ((𝐹‘𝐴) ⊆ (𝐹‘suc 𝐴) ∧ (𝐹‘𝐴) ≠ (𝐹‘suc 𝐴))) | |
10 | 8, 9 | syl6ibr 251 | 1 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (𝐴 ∈ ω → (𝐹‘𝐴) ⊊ (𝐹‘suc 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 {crab 3068 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 ⊊ wpss 3888 ∅c0 4256 ∪ cuni 4839 ↦ cmpt 5157 ↾ cres 5591 suc csuc 6268 ‘cfv 6433 ωcom 7712 reccrdg 8240 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 ax-reg 9351 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-om 7713 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 |
This theorem is referenced by: inf3lem5 9390 |
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