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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 9570 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 6843 | . 2 ⊢ (𝐹‘∅) = ((rec(𝐺, ∅) ↾ ω)‘∅) |
3 | 0ex 5264 | . . 3 ⊢ ∅ ∈ V | |
4 | fr0g 8381 | . . 3 ⊢ (∅ ∈ V → ((rec(𝐺, ∅) ↾ ω)‘∅) = ∅) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ((rec(𝐺, ∅) ↾ ω)‘∅) = ∅ |
6 | 2, 5 | eqtri 2764 | 1 ⊢ (𝐹‘∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 {crab 3407 Vcvv 3445 ∩ cin 3909 ⊆ wss 3910 ∅c0 4282 ↦ cmpt 5188 ↾ cres 5635 ‘cfv 6496 ωcom 7801 reccrdg 8354 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-om 7802 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 |
This theorem is referenced by: inf3lemd 9562 inf3lem1 9563 inf3lem2 9564 |
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