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Mirrors > Home > MPE Home > Th. List > inf3lemb | Structured version Visualization version GIF version |
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9629 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 |
---|---|
inf3lemb | ⊢ (𝐹‘∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inf3lem.2 | . . 3 ⊢ 𝐹 = (rec(𝐺, ∅) ↾ ω) | |
2 | 1 | fveq1i 6885 | . 2 ⊢ (𝐹‘∅) = ((rec(𝐺, ∅) ↾ ω)‘∅) |
3 | 0ex 5300 | . . 3 ⊢ ∅ ∈ V | |
4 | fr0g 8434 | . . 3 ⊢ (∅ ∈ V → ((rec(𝐺, ∅) ↾ ω)‘∅) = ∅) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ((rec(𝐺, ∅) ↾ ω)‘∅) = ∅ |
6 | 2, 5 | eqtri 2754 | 1 ⊢ (𝐹‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 {crab 3426 Vcvv 3468 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 ↦ cmpt 5224 ↾ cres 5671 ‘cfv 6536 ωcom 7851 reccrdg 8407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 |
This theorem is referenced by: inf3lemd 9621 inf3lem1 9622 inf3lem2 9623 |
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